Evelyn Lamb: Hello, My Favorite Theorem listeners. This is Evelyn. Before we get to the episode, I wanted to let you know about a very special live virtual My Favorite Theorem taping. If you are listening to this episode before July 16, 2020, you’re in luck because you can join us. We will be recording an episode of the podcast on July 16 at 4 pm Eastern time as part of the Talk Math With Your Friends virtual seminar. Join us and our guest Annalisa Crannell to gush over triangles and Desargues’s theorem. You can find information about how to join us on the My Favorite Theorem twitter timeline, on the show notes for this episode at kpknudson.com, or go straight to the source: sites.google.com/southalabama.edu/tmwyf. That is, of course, for “talk math with your friends.” We hope to see you there!

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Hello and welcome to my favorite theorem, the podcasts that will not give you coronavirus…like every podcast because they are podcasts. Just don't listen to it within six feet of anybody, and you'll be safe. I'm one of your hosts, Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City, Utah. And this is your other host.

Kevin Knudson: Hi. I’m Kevin Knudson, professor of mathematics at the University of Florida. So if our listeners haven't figured out by now, we are recording this during peak COVID-19…I don’t want to use hysteria, but concern.

EL: Yeah, well, we'll see if it’s peak concern or not. I feel like I could be more concerned.

KK: I’m not personally that concerned, but being chair of a large department where the provost has suddenly said, “Yeah, you should think about getting all of your courses online.” Like all 8000 students taking our courses could be online anytime now… It's been a busy day for me. So I'm happy to be able to talk math a little bit.

EL: Yeah, you know, normally my job where I work by myself in my basement all day would be perfect for this, but I do have some international travel plans. So we'll see what happens with that.

KK: Good luck.

EL: But luckily, it does not impact video conferencing.

KK: That’s right.

EL: So yeah, we are very happy today to be chatting with Belin Tsinnajinnie. Hi, will you introduce yourself?

Belin Tsinnajinnie: Yes, hi. Yá’át’ééh. Shí éí Belin Tsinnajinnie yinishyé. Filipino nishłį́. Táchii’nii báshishchíín. Filipino dashicheii. Tsi'naajínii dashinalí. Hi, everyone. Hi, Evelyn. Hi, Kevin. My name is Belin Tsinnajinnie. I'm a full time faculty professor of mathematics at Santa Fe Community College in Santa Fe, New Mexico. I’m really excited to join you for today's podcast.

EL: Yeah, I'm always excited to talk with someone else in the mountain time zone because it's like, one less time zone conversion I have to do. We're the smallest, I mean, I guess the least populated of the four major US time zones, and so it's a little rare.

BT: Rare for the best timezone.

EL: Yeah, most elevated timezone, probably. Yeah, Santa Fe is just beautiful. I'm sure it's wonderful this time of year. I've only been there in the fall.

BT: Yeah, we're transitioning from our cold weather to weather where we can start using our sweaters and shorts if we want to. We're very excited for the warmer weather we had. We're always monitoring the snowfall that we get, and we had an okay to decent snowfall, and it was cold enough that we're looking forward to warm months now.

EL: Yeah, Salt Lake is kind of the same. We had kind of a warm February, but we had a few big snow dumps earlier. So tell us a little bit about yourself. Like, where are you from? How did you get here?

BT: Yeah. I am Navajo and Filipino. I introduced myself with the traditional greeting. My mother is Filipino, my father is Navajo, and I grew up here in New Mexico, in Na’Neelzhiin, New Mexico, which is over the Jemez mountains here in Santa Fe. I went to high school, elementary school, college here in New Mexico. I went to high school here in Santa Fe. I got my undergraduate degree from the University of New Mexico, and I ventured all the way out over to the next state over, to University of Arizona, to get my graduate degree. While I was over there, I got married and started a family with my wife. We’re both from New Mexico, and one of our biggest goals and dreams was to come back to New Mexico and live here and raise our families where our families are from and where we're from. And when the opportunity presented itself to take a position at the Institute of American Indian Arts here in Santa Fe, it's a tribal college serving indigenous communities from all over the all over the nation and North America, I wanted to take that. I feel very blessed to have been able to work for eight years at a tribal college. And then an opportunity came to serve a broader Santa Fe, New Mexico community, where I also serve communities that are near and dear to my heart, where I've been here for over 30 years. And I'm really excited to have this opportunity to serve my community in a community college setting.

So, going into academia, and going into mathematics, it's not necessarily a typical track that a lot of people have opportunities to take on, but I feel very blessed to be doing math that I love serving communities that I love, and being able to raise my families around the communities that I love to. So I feel like you have a special kind of buy-in by engaging in a career that serves my communities and communities that are going to raise my families as well, too.

KK: That’s great.

EL: Nice. So I see over your shoulder a little bit of a Sierpinski triangle. Is that related to the kind of math you like to think about? Or is it just pretty?

BT: Yeah. One, it’s pretty. When I was at the Institute of American Indian arts, most of the students there, they're there for art. They come from Native communities, and they're not there to do mathematics, necessarily. So part of my excitement was to think about ways to broaden the ideas of mathematics and to build off of their creative strengths. And that piece is a piece that one of my students did. They did their own take on a Sierpinski triangle. I have a few of those items from my office where they integrated visual arts and integrated creative aspects of mathematics from cultural aspects as well, too.

KK: So I always think of Native American artists being kind of geometric in nature. It feels that way to me, I mean, at least the limited bit that I've seen. Is that sort of generally true?

BT: The thing about Native art is that Native cultures are diverse in and of themselves too. So there are over 500 federally-recognized tribes, and in Mexico are over 20 tribes alone, 20 nations alone, and each of them have their own notions of geometry and their own notions of their kinds of mathematics that they engage in with respect to the place that their cultures, their identities, and their languages are rooted in. So, yeah, a lot of it is visual, and geometric, because that's what we see. But there's also many I imagine that we don't see, that's embedded in the languages and the practices. Part of my curiosity is seeing how we can recognize what we do and what our traditions are, how we can recognize that as mathematical. And it might be mathematical in the sense that we, as professional mathematicians, might not be accustomed to seeing or experiencing. And, you know, I'm still trying to understand my own cultures, languages and traditions too. So I know mathematics more than a lot of how I experience my own culture. So on one hand, I'm seeing things from a traditional mathematician brought through academia, but I’m also trying to understand things through the lens of someone who's trying to better understand my cultures and histories.

EL: So what is your favorite theorem?

BT: The theorem I chose today was Arrow’s impossibility theorem.

KK: Nice.

EL: Great. And this will be a timely one, at least for the US, because it will be airing—I mean, I guess the past two years basically have been part of the 2020 presidential season—but really in the thick of it. So yeah, tell us a little bit about what this is.

BT: So I'll say more about why I'm kind of drawn to this theorem. So it's a theorem that basically says that there is no perfect ranked voting system, or no perfect way of choosing a winner and, by extension, for me, it kind of brings up conversations about how democracy itself isn’t perfect and that it's really hard to say that a democratic system can accurately represent the will of the people. And I was drawn to this theorem because as I started thinking about the cultural aspects of mathematics and mathematics education, I'm also interested in the power dynamics and the political dynamics and the sociopolitical aspects of mathematics and math education. And a lot of what's out there and written about math education talks about using quantitative reasoning and quantitative analysis and statistical analysis to really engage in critical dialogues and examining inequities and injustices in the world. And all of that is rich and engaging and needed and necessary ways that we can use mathematics to view the world. But the mathematician part of me still misses the definition-proof-lemma aspect of engaging in mathematics. So this theorem kind of represents a way of engaging in politics through some of the theorem-definition- lemma aspects of it. So the way that I understand Arrow’s theorem, and I mentioned this to you before, that I don't know the ins and outs of this theorem, I just really like the ramifications of it and the discussions that it generates. But it basically starts with the idea that we can describe functions where we're considering a way of choosing a winner of an election from a list of candidates. And we're taking each voter’s ranked preference of those candidates. So one thing that we're assuming is that each voter can rank a list of n candidates, A1 through An, and if everyone can rank their preferences, then a voting system would be a way to take all of those, those ranks, or those ballots, and choosing an overall ranking that is supposed to indicate an overall preference for the group of voters.

And what Arrow’s impossibility theorem talks about is that we want values, and want to describe good ways of what a good voting system is. So we want to describe list of criteria that shows that we have a good voting system. So the list of criteria that involves Arrow’s impossibility theorem talks about 1) and unrestricted domain; 2) social ordering; 3) weak Pareto or unanimity; 4) a non-dictatorship; and 5) independence of irrelevant alternatives. And I'll go through what each one means. So basically, an unrestricted domain means that we want a voting system or a way of choosing a winner to be able to take any set of ballots with any number of candidates and be able to give some overall ordering, that these functions are well-defined. So the unanimity condition talks about if everyone prefers one candidate over another, where every single voter has one candidate ranked over another candidate, then the overall function that turns the ballots into an overall social ordering should indicate that that candidate is preferred over the other candidate. And we also don't want a dictatorship, right? And the idea of that mathematically defined is that we don't want one voter deciding exclusively what the overall social ordering is of the candidates. And so we don't want a dictatorship. And we want an independence of irrelevant alternatives, and what that what a lot of people think about as an example of is a “spoiler” candidate or a third party candidate, where even if everyone prefers one candidate over another, that a change in order of a third or other candidate, without disrupting that other order, shouldn't change the overall outcome of an election. They relate that to how sometimes third party candidates can be a spoiler for an election even though overall, it looks like a plurality of voters might prefer one candidate over another. But certain voting systems can have that characteristic where third or other other set of candidates can disrupt the outcome of that election.

KK: I’ve never heard of that.

EL: Wouldn’t it be terrible if that ever happened? [Note: These statements were delivered somewhat sarcastically, presumably referring to the 2000 Presidential election in the US]

BT: Right, right, right. So what Arrow’s impossibility theorem says is that those all may be desired characteristics of a voting system or a social choice function, but that it's impossible to have all of those criteria in a voting system. So the general outline of the proof is that if we have a system that has the unanimity criterion, and an independence of irrelevant alternatives, that if we have those two criteria in a social choice function, then the voting system must be a dictatorship. So if we add those assumptions, then we can go through and show that there is a voter whose sole ordering determines the overall ordering of the voting group, of the voters.

KK: That’s how I always learned this theorem, is that you set down these minimal criteria, and the only thing that works as a dictatorship, right?

BT: Right.

KK: These criteria are completely reasonable, right?

EL: You can’t have it all.

BT: Right, right. They're not outlandish. They're what we might think of as things that we might value in a democracy. And, of course, these, these things don't perfectly replicate what's going on in the real world, but the outcome is still fascinating to me that mathematically, we can show that we can’t have all these sets of what we think are reasonable criteria in a voting system.

KK: Recently, maybe in the last two years, I’ve been getting interested in gerrymandering questions. And there's there's a similar sort of theorem that got proved in the last year or two, which essentially says that, you know, people don't like these sort of weird-shaped districts, they think that's bad somehow, because it's on unpleasing to the eye. But apparently — and there’s also this idea of the efficiency gap, where you sort of want to minimize wastage. So if you laid out some simple criteria, like you want compact districts, and you want to make the efficiency gap, minimized that, then the theorem is you have to have weird shape districts, right? So it’s sort of an impossibility theorem in that way too. So these these kinds of ideas propagate through all of these these kinds of systems,

EL: The real world is impossible.

BT: Right. And even by extension, you know, in many voting theory classes, there's a districting problem, which relates to a good metric for measuring compactness. But then the apportionment issue as well, that it's very hard, if not impossible, to find a fair way of apportioning a whole number of representatives that's proportionate to the state's population, relative to the overall population of the country.

KK: Yeah.

BT: And so yeah, this is one of my favorite theorems because it kind of opens the door to those conversations and gives me another way of thinking about when representatives, or people who talk about the outcomes of elections, say things like “the people have spoken,” “this is the will of the people,” “we have a mandate now,” that I think these outcomes really complicate those claims and should really give us a critical eye and a critical way of really discussing what the will of the people is, and how those discourses really perpetuate the idea that voting, and voting alone, can accurately indicate the will of the people and that that's to be accepted, and that we move forward with them.

EL: Yeah. So have you gotten to use these Arrow’s paradox or any of these other things in classes?

BT: When I was at the Institute of American Indian Arts, I tried to develop a voting theory class. And we got into that and talked about that. And it interested me too because the voting system on the Navajo Nation, we vote for our own council and our own presidents too, and I use this as a way to think about how we have a certain candidate in Navajo Nation who's always running and is seemingly unpopular. And the voting system for president in Navajo Nation is that we have that two-party runoff system where we vote for our top choices and that the top two vote getters participate in a general runoff election. And for a few consecutive elections, this one candidate that is seemingly unpopular just gets enough votes to get into the top two for the runoff election and then gets overwhelmingly outvoted in the general election. So I think for me it was a fascinating way to engage in these kind of mathematical ideas, or mathematical discourses, while talking about some of the real outcomes that are going on in our nations, in our communities, in our efforts towards our self-determination and sovereignty. So I wanted to tie in something that's mathematical, where we can talk about mathematical discussions, with issues that are contemporary and real to our, our peoples.

EL: It’s something I always wonder about is, you know, we've got a theorem that says voting is impossible — or it says that, you know, it's impossible to actually say, like, this is the will of the people. But do you know if much research has been done about, like, real sets of choices that people have and what voting systems might be — do they really experience this paradox, or in the real world, do they have these strange orders of preferences that that confound ranked choice voting rarely?

BT: I imagine that there is research out there and there are people who have engaged in it much more than I have. But something that makes me curious are some of the underlying assumptions that go into Arrow’s theorem and what has been mathematized as necessary criteria, and the values that those might be representative of for certain groups of people. For example, I guess you could call it an axiom of many these voting theory theorems in mathematics is that one voter is one vote, and you know, there are systems where that might not be true. But one of your criteria is one person, one vote. And that one person votes for their own interests and their own interest only, and there are extensions of these criteria where if we have other non-ranked voting systems, then it can help.

But let me backtrack: one of the outcomes of Arrow’s theorem is that when people know that it's impossible for the outcome to really represent the will of the people, then it could result in people voting for candidates other than their first option because they know that voting for someone other than their true option because we election in favor of something that's not of their desire. So we have people voting against their own actual first choices. And that happens with ranked-choice voting, and some of the extensions of these conversations have been about voting systems that don't require ranked choice. So perhaps giving each candidate a rating, and it helps alleviate some of those issues with ranked-choice voting, and it helps alleviate those issues of third-party candidates, where you can still give your candidate five stars out of five, like an Amazon review, but still really give perhaps a better indication of your true view of the candidates, rather than a linear ranking. So it kind of reveals that there are some issues with just linear ranking of candidates, when the way that we think about in value and understand our preference of candidates might be much more complex than a simple 1 through n ranking. But kind of going back to what I think this could mean for communities and other societal perspectives, is in many democracies, that one vote-one choice is kind of an assumption that that's what we want. But for many communities, perhaps we want to vote for something that does benefit an overall view of the people. What would that look like as a criteria if we allowed for something like that? What would we do if we allow criteria, or embedded in our definitions, some way of evaluating how if when we register a vote, that we're all not only taking into account our own individual interests, but the interests of our land, of our communities, of our nations. So those are cultural values that are not assumed in the current conversations, but for many communities in many Indigenous nations, those are some things that are real and necessary to think about. What would that look like if we expand those and then be critical of those assumptions that are underlying these current conversations on voting theory in mathematics.

EL: So one of the other things we do on this podcast is We ask our guests to pair their theorem with something. What have you chosen to pair with this theorem?

BT: I have a ranking of three pairings.

EL: Great. I’m so glad! Excellent.

BT: So I have 1-2-3. So I'll give my third choice first. The third out of three pairings: green chili cheeseburgers.

EL: Okay.

BT: And in New Mexico, everyone has their favorite place to get a green chili cheeseburger, and we take pride in our green chili, and every year any contest about the green chili cheeseburger and who has the best green chili cheeseburger causes some conversation, and it causes some controversy and rich discussions over who has the best green chili cheeseburger. So, I think about that as a food that has a lot of controversy as to who has the best green chili cheeseburgers in New Mexico. The second pairing is another food item, the Navajo taco.

EL: Oh yeah. Those are good.

KK: What’s in those?

BT: So, well, what we call a Navajo taco is a piece of frybread with toppings often involving meat and cheese, with lettuce and tomato and maybe some chili. And this is another controversial discussion in Native communities because we call it a Navajo taco, but it's not just Navajos who make this kind of dish, because many communities make their own versions of frybread. And so some places call it Indian tacos, and there's a lot of controversy over which community first introduced the Navajo taco and why some people call it the Navajo taco and others call it Indian tacos. And so in Native communities, there's a lot of controversy over what constitutes the best version of this dish. And the other reason I'm pairing that is the frybread itself comes from a time where it was created out of necessity for survival, where the flour that had been rationed out to our communities was rancid, and in order to actually make it edible, it was deep fried. And so on one hand, it represents a point in time where our communities were just fighting for survival, and it also represents their ingenuity, and became a part of our everyday practice. But at the same time, it's a reminder that that was something that was imposed on our communities, much like voting systems nowadays. It's an act of our survival and our sovereignty, the voting systems that we have in place. But I think there's also need to come back and have other conversations about what's good for our communities.

And the first-ranked pairing is mathematics itself with Arrow’s theorem. So we have a lot of conversations about how mathematics is universal, mathematics is for everyone, that everyone can do mathematics, and that everyone can participate in mathematics. But for many people from from equity, justice and diversity perspectives, we want to be critical about who has access to mathematics, whose ideas of mathematics are represented in our mainstream ways of thinking about mathematics. Just like we think about democracy as being the will of the people and being a representation of all the people, that Arrow’s is kind of a critique of that notion of democracy. And I think mathematics, we can take a lesson from this theorem and think about what we mean when we say mathematics is universal or mathematics is for everyone or mathematics is for all, when this term itself is kind of a democratic take on mathematics, that everyone can do mathematics, and everyone can be an equal participant in mathematics. But, you know, we think the same thing about democracy, and this theorem says that there are some issues with that. So I'm interested in seeing how we can take this lesson and how we can think about how we can be more critical about the ways we think about mathematics itself.

EL: Yeah, well, you know, Arrow’s paradox is not about this, but we have issues with people who can't vote for various reasons and should be able to vote, or places that shut down polling places in certain communities to make it so people have to stand in line for six hours. Which is, you know, not easy to do if you've got a job that you need to get to. So yeah, there's so much richness. I love that you paired a ranking of three things with this. And now I feel like we should also vote on these, but I just don't think it's fair for one of them to be math. I mean, you’ve got two mathematicians here, three mathematicians here in total. I think it's going to be a blowout.

KK: No, tacos win every time, don’t they?

EL: I should have known.

KK: This is a really good pairing. I like this a lot.

EL: Yeah.

KK: We also like to give our guests a chance if they want to plug anything. Where can we find you online for example, or can we?

BT: Probably the best way to find me is on Twitter. My Twitter handle is @lobowithacause.

EL: Yeah. You'll see him popping up everywhere. Is that the mascot for the University of New Mexico?

KK: It is, the lobos.

EL: And I believe a talk that you gave at the Joint Math Meetings, is there video of that available somewhere?

BT: I was told that there would be video. I haven't found it yet. There was a video recorded. And I'll follow up with that and see that it gets out. I'll make an announcement on Twitter.

KK: I’ve noticed those have been trickling out kind of slowly. It'll show up, I think.

EL: Yeah, we'll try to dig it up by the time we put the show notes together so people can watch that. Unfortunately, I was still making my way to Denver when that happened, so I didn't get to see it. So selfishly I very much want to see it. I heard really good things about it. So thank you so much for coming on here and giving us a lot to think about.

BT: Oh, it was an honor. And you know, I love your podcasts.

KK: Thanks so much.

BT: I love what you’re doing. I had fun in listening to your other podcasts in preparation for this and loved hearing Henry Fowler and shout out to Moon Duchin too. I heard that you, Kevin, went to that gerrymandering work in Boston a few years ago. I was there too. And I had a great week there.

EL: Oh, nice.

KK: That was a big workshop. There was no way to meet everybody. Yeah,

EL: Thanks for joining us, and have a good rest of your day.

BT: Thank you. Thank you. You too.

In this episode of the podcast, we were happy to talk with Belin Tsinnajinnie, a professor at Santa Fe Community College, about Arrow's impossibility theorem, which basically says that a perfect voting system is impossible. Below are some links you might enjoy as you listen to the episode.
Arrow's impossibility theorem
Cardinal voting, an alternative to voting systems that are based on ranking the options
Our episode with Henry Fowler, who was at the time on the faculty of Diné College and is now at Navajo Technical University
Our episode with Moon Duchin, who studies gerrymandering, among other things
Belin Tsinnajinnie on Twitter