Episode 133 - What Is the Point of Pointless Topology?
Mathematics is a vast field that explores the different dimensions of the world. It broadens our perspectives on the infinite possibilities of what we can do. One branch of mathematics, topology, deals with studying preserved properties from deformations. What are its uses, and how can we apply it in our everyday life?
This episode is about the field of mathematics known as topology. Max and Aaron use common language to explain the phenomenon using real-life examples. They also dwell on an interesting subfield, pointless topology. They also talk about the similarities of the field with knot theory. Finally, Max and Aaron share their Metaculus predictions on the upcoming US presidential elections.
Tune in to the episode to learn more about the application of topology.
Here are three reasons why you should listen to the full episode:
Discover the applications of topology in the real world.
What are pointless topology and topological spaces?
Find out about Max and Aaron’s predictions on the US presidential elections.
Resources
Techmeme Ride Home Podcast – Stay up-to-date on technology news every day in just 15 to 20 minutes.
Intro to Topology by Hotel Infinity
A Hole in a Hole in a Hole by Cliff Stoll on Numberphile
What is a Knot? by Numberphile
Topology vs “a” Topology by Tai-Danae Bradley on PBS Infinite Series
Course Lectures
From Frederic P. Schuller’s 24-part lecture series, “A thorough introduction to the theory of general relativity”
Advanced Resources
These are fairly advanced, and I’m interested in recommendations for more introductory books.
Introduction to Topology by Bert Mendelson
Get Topology: A Categorical Approach for FREE at The MIT Press!
Topology via Logic by Steven Vickers
Frames and Locales: Topology without Points by Forge Picado
The point of pointless Topology (1983) by Peter Johnstone
Related Episodes
Episode 114 about presidential candidates getting involved in podcasting
Episode 85 with Anthony Aguirre about the Metaculus prediction engine
Episode 84 with Anthony Aguirre about the true nature of space
Episode 4 on hill climbing and the algorithm for cracking the substitution cypher
Episode 57 on nearest neighbor algorithms
Episode Highlights
Presidential Candidates & the Podaverse
Joe Biden has started a podcast, which he hasn’t updated since May 11.
Trump doesn’t strike Aaron as a podcast kind of guy. He’d go for tweets or videos.
Topology vs. Topography
A topological map is a map with rings that shows elevation.
Topology and topography has some base-level connection, but the latter is in the field of geography.
What Is Topology?
Max discovered it as an undergrad and circled back to it in 2015.
The textbook definition of topology is studying certain types of spaces and how they’re connected to each other.
Max thinks it's abstract, but it has some relationship to the real world.
You can apply it to physical objects as well as non-physical ones, such as the hypothesis space.
The big idea in topology is continuity. Distances don't matter anymore, only shape and form.
The Concept of Neighborhood
In real-world situations, you can put it as the connections that matter.
One key term in topology is “neighborhood,” which refers to the points around you. It's like moving to a new city and having the same neighbors as before. Hence, you're in the same situation.
Listen to the full episode to hear Max’s example on how topologists classify surfaces.
Topological Spaces
If your hypothesis space is a topological space, then that defines where your neighbors are. That is the most general kind of space in which you can have a hill-climbing algorithm.
A vector space usually has distances and is more structured than a topological space.
A ring topology is when workstations are connected and form a circle with continuous flow.
A set that surrounds a point is called a neighborhood. The point is not required to be the center of the neighborhood.
To Aaron, it brings to mind the Heisenberg uncertainty principle, in which you can know you're inside of a group but you don't know where you are inside that.
The Concept of Continuity
Continuity is such a difficult concept because you're not talking about going from one point to another.
In terms of limits in calculus, you can tell what’s happening at a point as long as you’re in its neighborhood.
Looking at a continuous analog gives you more insights that you wouldn't necessarily have otherwise. It allows interpolation and extrapolation in a way that the discrete model doesn't.
Open Sets
Most topology is not pointless. Pointless topology, however, doesn't take points as the main structure.
An open set in topology refers to a neighborhood of all its points. You could go in any direction inside an interior and still be in it because it has no boundary.
Open sets are the fundamental unit defining points. A point is the collection of open sets that contain it.
The idea is to consider an area of a certain extent, not points. It is more accurate in describing the real world because any measurement that you take has error bounds.
Listen to the full episode for Aaron’s example on Venn diagrams.
Finding an Overlap
With open sets, you have a set union and a set intersection.
A union is combining two sets and getting something bigger out of it.
An intersection is an overlap or commonality.
A topological space of open sets forms a lattice.
The Point of Pointless Topology
It's an interesting way to look at problems and connections while setting aside notions of distance and form.
In some ways, it can eliminate bias.
Form in Pointless Topology
It’s all about connections, not distance.
The form is the knowledge of the connections that define the possible shapes a space could take on.
Social network theory is highly related to the discrete side of topology.
Topological Transformations & Knot Theory
If you take a string, tie it, and glue the ends together, can you make them look similar? Knot theory answers that question.
The two types of topological transformations are either keeping all the neighbors or continuously deforming them.
Surviving the COVID-19 Pandemic
Even if the number gets to zero, people will still be crazy about it.
Remote working is here to stay.
Max and Aaron share the struggle of doing tasks at home. They say the podcast doesn't feel like work because they're not filling out a time card.
US Presidential Elections Predictions
The DNC is happening this week.
Max and Aaron share their Metaculus predictions on the presidential elections.
They think it’s at least 90% likely that Biden will debate Trump.
5 Powerful Quotes from This Episode
“I think it's (topology) totally abstract, like a number is abstract or real numbers are abstract, but obviously it has some relationship to the real world. Otherwise, you wouldn't be studying it.”
“You can't consider points. You actually have to have a little substance in there, which, if you think about it, is, actually, in some ways, more like closer to describing the real world because any measurement that you take has some error bounds around it.”
“I'm envisioning this pointless topology approach, if I want to describe the equivalent of a point there, is that I just need to keep drawing more and more circles on that Venn diagram so that I can narrow and narrow and further narrow the section right in the middle where all these circles overlap and try and more clearly define and constrain that ever-shrinking space in the center.”
“I just think it is an interesting way of looking at problems and looking at the world where you're just thinking about the connections between things, and you're kind of setting aside your notions of distance and form. And then in some ways, it's kind of a different way to approach problems, and maybe it's a way where you can kind of look for bias or eliminate bias.”
“We have the connections, but we don't know what they mean yet, and so let's see what the connections look like. And then on top of that, we can sort of give them weight after just looking at the connections themselves.”
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To expanding perspectives,
Max
Transcript
Max Sklar: You're listening to The Local Maximum Episode 133.
Time to expand your perspective. Welcome to The Local Maximum. Now here's your host, Max Sklar.
Max: Alright, everyone, you have reached another Local Maximum. Welcome to the show. Today, I'm joined by Aaron. How are you doing, Aaron?
Aaron: I'm doing well. It's another summer night on the East Coast here.
Max: Yeah, I know. It feels kind of warm for some reason. And, you know what, when was the last time? I think it feels like we just spoke recently, but I feel like it's been over a month since we've done a podcast together, I mean. Well, we recorded two at once. That's why. You know, so it’s...
Aaron: Doubleheader throws things off a little.
Max: Yeah, we should do more doubleheaders. It's not gonna be a doubleheader today. Today, we're gonna try something a little new. Because I've always been fascinated, you know, I'm into machine learning and all that and Bayesian inference—I talk about it all the time on here. For some reason, I'm interested in a mathematical field called topology. And I say, for some reason, maybe we could figure out why I'm so interested in it, and I have some ideas, but maybe—I haven't really explained this to anyone else yet, so we're going to try to have a conversation and see if we can put the pieces together. So this is not going to be kind of a course material yet, but let's just have a conversation about it. Before we start, let's see, let's talk about what's going on with the Local Maximum and with the news in general. But one news item that I wanted to follow up on, one of my favorite episodes that we've done previously, was 114. That was the one with smart toilets. You remember that one?
Aaron: How could I forget?
Max: Yes, how could I? Well, actually, I went through all of the episodes recently, and I painstakingly wrote URL slugs for them for SEO. Let me tell you, I think I remembered most of them. There might have been a few episodes that I wouldn't have been able to recall but very interesting, I kind of remember what they're all about. But anyway, that when we talked about Joe Biden joining the podaverse, and I was kind of excited about that. The presidential candidates were having their own podcasts and going on podcasts, and I thought maybe I can't get them on my show, but, you know, as you can always do, maybe you can get a guest of theirs also on your show. That's always possible, or like, you know, if you have a small show, you can do that. But, man, it turned out, so I thought tonight's the night of the DNC, it turns out he hasn't updated his podcast since May 11. So it's been a while.
Aaron: I know we talked a couple weeks ago, and you mentioned that it had gone dormant. I thought, right now, you were going to tell me that they've revived it. We're going to be coming out with new, you know, Kamala and the Joe episodes to revitalize by the fresh blood, but…
Max: I don't think…
Aaron: Maybe they will, but they haven't announced it yet, apparently.
Max: Yeah, I don't think this is the year it's gonna happen. I think they realized this is hard work.
Aaron: And on the other side, Trump doesn't strike me as a podcast kind of guy.
Max: No.
Aaron: I think he'd go straight for video. It's either tweets or video. Podcast is a middle ground that he sees no use for.
Max: Yeah. I mean, you could probably put him on a microphone and see what happens. That would get a lot of listens, but they haven't done it yet.
Aaron: Well, and apparently, he calls into what is it, like Fox and friends, or at least it used to, quite differently.
Max: Sure.
Aaron: It's a little bit different. Anyway.
Max: I feel like, for him, that's just like interactive TV.
Aaron: Well, what we really need to do is set up a column number. So if there are any presidential candidates or their running mates who want to call into the Local Maximum, you just dial 555-5555.
Max: Alright. I think the folks are gonna get annoyed that we're not getting to the point here, and also, this is not a, we're making this very non-evergreen by talking about this. So maybe we can move on. I know it got to...
Aaron: For those of you in the future, I hope everything's better, and all the problems have been solved. So let's now move on to something that has nothing to do with politics.
Max: I'm sure it is. I'm sure it is. Okay, so first of all, before we get into the questions, like what did you know about topology before going into this, Aaron?
Aaron: When you say topology, I think of topo maps, which I'm sure that there's some connection to that.
Max: Is that like...I don't know what topo maps are.
Aaron: Oh, yeah, the topological map. So they’re the maps with the rings that show you what the elevation is, which, I'm sure, has some base level connection here. But you're not talking about, you know, the geography side of things.
Max: Right.
Aaron: My impression is you're much more on the theoretical mathematics side here.
Max: Right, right, right. So I think that's called topography versus topology.
Aaron: Oh, you might be right.
Max: Yeah. But I'm not this.
Aaron: This is the problem with shortening things.
Max: No. I mean, it's a good distinction. Alright. Cool. We can get started, and let's see if we can figure out what this thing is.
Aaron: Well, actually, I want to deviate on the basics.
Max: Well, also we’re going to talk about pointless topology. Yeah. Okay.
Aaron: So you mentioned before, you know, it's unclear why you're so particularly interested in this, but just when did you first get interested in topology? Does this predate your interest in machine learning?
Max: I think it does. And by the way, we're trying to figure out whether it's pointless or not, and in a sense, it is. Because at the end, we're going to talk about an even lesser-known field called pointless topology, which is one of the more mind-blowing things. So, yes, I feel like I had read a little bit about it as an undergrad, and I tried to get into it, but I didn't quite grasp what these math books were trying to say about it. And now, I feel like I have a much better grasp of it and so I could actually explain it to the folks here in the Local Maximum, you, the listener. But yeah, I would say that...so, when did I get re-interested in it would be a good question.
Aaron: Was this something that really clicked like during grad school? Or was this more you were exploring it on your own?
Max: On my own. Probably in like 2015.
Aaron: Gotcha.
Max: And now, I've gotten circled back to it.
Aaron: So that's a little bit of background there, but let's get back to the fundamentals here.
Max: Sure.
Aaron: What is topology? I'm gonna throw out the basic, most open-ended question I could think of.
Max: Yeah. Well, that's a good question because there's no obvious answer to it. And I looked at some of my math textbooks here, and they don't really do a very good job of defining it. They say, “Introduction to topology. Alright, here's your first definition.” And it's like, “Wait a minute. What is this thing?” So as far as I can tell, it's the study of certain types of mathematical objects, certain types of spaces. And so, the spaces that we are most familiar with on the Local Maximum would be, you know, what I would call hypothesis spaces. Like, “Hey, I have a bunch of things that I'm not sure whether they're true or not, and I'm gonna like arrange them somehow,” or, you know, you could think of like Euclidean space. I'm sure, I don't have to tell you, I know you went to MIT, you know Euclidean spaces, but like how would we describe that? Let's say, like three-dimensional space, so it's kind of like the study of how space is connected, how locations are connected to each other.
Aaron: So I was gonna ask, is this physical, or is it something more abstract? And it sounds like the answer is both, or it can be either.
Max: I think it's totally abstract, like a number is abstract or real numbers are abstract, but obviously, it has some relationship to the real world. Otherwise, you wouldn't be studying it.
Aaron: Sure, sure. You could use it for like physical objects in spaces but also for some of these more abstract stuff, we're putting stuff in a matrix of data points and how they relate to each other as well even if that's not their physical relationship. It's something like, is it more abstract.
Max: Oh, yeah. It could be used, you know, you could be talking about something that, it is where you're thinking of a physical object—and I'm sure it can apply to areas where you're not talking about physical objects. Like a hypothesis space is not a physical object.
Aaron: Yeah. You can only stroll through a hypothesis space in the most abstract of fashions.
Max: Right. Exactly, exactly. So, you know some of the big ideas in topology are continuity, like it looks at, you know, “Hey, if you have a curve, there's like some continuity there. You're not like lifting up your pen,” as they'll often say in math class in high school, but, you know, this kind of formalizes it a little bit. And the thing about topology is that distances don't matter anymore; it’s only shape and form that matters. So I wrote here like a football equals a basketball. The reason why a football is equals a basketball in topology—the idea is this—like, okay, you take a football and you kind of blow it up, and you kind of push the sides in, so it's like it's round, and then you kind of expand it like a basketball. And if I'm standing, if I have a point somewhere on that football, and then I look at the points around me before the transformation, and then I look at the points around me after the transformation, they’re the same points
Aaron: So the distances could change, but you've already said that distances are irrelevant. So it's, you know…
Max: Right.
Aaron: If you're a node and what nodes are you connected to is what matters.
Max: Exactly. I mean, there are a lot of—and that's a good way of putting it—like there are a lot of real-world situations where it's the connections that matter. And you know, the way I say it, like one key term in topology is neighborhood. So neighborhood is like the points that are around you. And so, you know, I say, “Hey, let's suppose we all live together in one city, and then one day something happens, and we all have to move to another city.” I couldn't imagine a situation where that would come, I guess a year ago, I could never imagine a situation that could. “Okay, and then we move to our new city,” and then “Hey, what do you know? All of your old neighbors are the same as your new neighbors.” That is a topological transformation, that is, it's like it's an isomorphism. You got the same neighbors as before, so you're kind of in the same situation. Even if, you know, the density is different. You know, let's say you have bigger plots of land or something like that, or your land’s now long and skinny rather than square.
Aaron: I can only dismiss, or I can only accept the fact that distances are irrelevant if I don't think about it too hard. So I'm going to shove that in the back of my head and try to move forward with this.
Max: Okay. Well, okay. I think it'll become more clear as time goes on, as we talk about this more.
Aaron: Okay, so we've got, you said, it's who your neighbors are, but what was the other feature? Was it the curves and…?
Max: So we'll get to this in a minute.
Aaron: The shapes?
Max: I want to talk...yeah, yeah. And so—oh, a good example here is like a doughnut. The surface of a doughnut is not equivalent to the surface of a ball or basketball because you can't transform a doughnut into—so this coffee mug, this mug that I have here with my tea, you can actually can transform the surface of this into a surface of a doughnut because it's got one hole in it. But you know, it would be a weird transformation, but you could do it. And so that's sort of how topologists classify surfaces.
Aaron: I'm envisioning something very trippy with the transformation of a coffee cup into a doughnut.
Max: Right? So you basically have to would—flatten it, and then you'd have to make the handle really big, and then that becomes the doughnut. Yeah.
Aaron: Oh, I'm thinking—okay, so we're talking about the hole between the handle and the body?
Max: Exactly.
Aaron: Not the hole into which you pour the liquid?
Max: No, no.
Aaron: Oh, okay,
Max: So if there's no handle, it's the same as a basketball.
Aaron: Okay, that clarifies something. I don't know if it's helpful, but it clarifies something.
Max: You’re basically, you’re punching the basketball, and you make a little indent, and that becomes like the, where the tea or coffee goes in the cup, and then you keep pulling it.
Aaron: Okay, I'm with you on that. Hopefully, those of you listening have excellent spatial memories, spatial visualization, and we have not completely lost you.
Max: Yeah.
Aaron: So, I have the most vague idea of what the concept is here.
Max: Okay.
Aaron: But why do we care, right? What is this good for? And you've said that there are obviously applications you can take this thoroughly abstract concept and use it for something, but what is that something?
Max: Yeah, so one thing that I'm thinking about in terms of machine learning, in terms of Bayesian inference is we've talked about the nearest neighbor algorithm before we've talked about—I don't remember which one was nearest neighbor—but I remember in Episode 4, we did that, where I cracked those codes that you generated. Do you remember that?
Aaron: Yep, yep.
Max: The substitution cipher. So the assumption there was that solutions that are nearby to the solution that I'm at are going to be similar, more or less. And that kind of allows us to do hill climbing. It's like, “Well, I have a solution right now. It might not be very good, but I look at the solutions around me, and then I jump to one that is better, and then I keep doing that. And then when I jump to one that's better, the ones around that are going to tend to be, I could probably find one that's better than that.” And you could keep doing that and doing that until you reach a local maximum until you get to what could be a good solution, or at least a lot better than you're starting with. That's hill-climbing.
And so, I think topology comes up with this is, if your hypothesis space is a topological space, then that defines where your neighbors are, and then that is the most general kind of space where you can have kind of a hill climbing algorithm in. And what's interesting about it is that distances—since distances don't matter, oftentimes. Like, okay, a lot of times in the nearest neighbor algorithms and machine learning, it's all about distance, right? I want to find the perfect analogy, so the most closely perfect analogies. I want to find the examples that are most close to the ones that I am trying to predict. So, let's use an example. Let's use the presidential election, right?
Aaron: Okay.
Max: We have this year's presidential election, and then we have how many examples of presidential elections? I don't know exactly how many, but I'm not gonna check.
Aaron: We’ve had 45 presidents.
Max: Yeah.
Aaron: But more elections than that.
Max: I'm not gonna try to do it in my head. I mean, well, there's 25 a century, so there's at least, I don't know, it’s 50 something. Okay, so you define some distance metric on the presidential elections, and you say, “Okay, what's the most similar one to this one? What's the best analogy?” And if you look at the pundits, they're always trying to come up with analogies. But the problem is, any distance metric that you come up with, you could say, “Hey, this is a bad distance metric. I have another one that's better. And that's why this year is more like this other year and not like the year that you were saying.” And so, when you just talk about the connections and stop thinking about the distances, that kind of allows you to think, “Okay, what distance metrics can I impose on this? And maybe we could try a few different ones.”
Aaron: Okay, so if…
Max: And by the way, I don't think, like, I haven't found machine learning researchers who talk about topological spaces in this way. This is just what I'm speculating.
Aaron: Well, is that a case of they don't talk about it in this term, but that's what's underlying the process, or...?
Max: Yeah, it could be, but…
Aaron: It’s an adjacency.
Max: It could be, but most of the time, they're just using Euclidean space. Or they're using, you know, something very a little less abstract, which might make sense, but...
Aaron: Okay. So how does this differ from, I mean, I guess, we said in the context of the k-nearest neighbor and that type of thing that—I assume we were working in more of a vector space type approach?
Max: Right?
Aaron: What separates this from that?
Max: So, a vector space usually has distances, and a vector space has more structure than a topological space. And as you know, you must have taken some really trippy calculus courses at MIT, am I wrong about that? Like they probably told you like, “Here's a vector space that somehow represents spaces, you know it, in the real world.” But they might have given you extra dimensions that might have been like, “Hey, you multiply it by this matrix and then it like stretches...”
Aaron: I never got into the nitty-gritty math stuff. I was multivariable calc and then differential equations, and then beyond that, I did a bunch of numerical methods stuff.
Max: Oh, my god.
Aaron: But I didn't go into like, I don't know, would that be leaning into the linear algebra direction?
Max: Yeah.
Aaron: Or…?
Max: Well, yes. So anyway, it just has a lot more structure in terms of how like the space is arranged, or topological space is more general. It's like, “No, these are the points that are—it's like this is how the space is connected to how the points connect together. And I don't care about distances, I don't care about angles, I don't care about,” you know, “a lot of things you might want to care about in engineering.”
Aaron: This is hurting my engineer brain because those are the only things that we care about.
Max: Right, right. But in some areas, specifically like in physics, shape becomes more important. So, yes. But oftentimes, you start with a topology, and then you could impose, as I said, like distance form on top of it.
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Aaron: Okay, so I guess these are all topological spaces we're talking about?
Max: Sure. Okay, so lets…
Aaron: And I think when I mentioned before having interconnected nodes that's more of maybe—I don't know if it's the right terminology—but I would think that would be kind of like a network topology.
Max: Exactly, exactly.
Aaron: So other than then possibly not having distance in there, what's the key difference?
Max: Yes. So that's actually a good example where distance doesn't matter. Like, let's suppose we are on a communication network or computer network, right?
Aaron: Okay.
Max: So that is a network topology. Let's say there are like three workstations, and they're all connected to each other. So that's like a triangle topology. I don't know if that's the technical term—I just made that up.
Aaron: I would call it a ring topology.
Max: Okay, yeah. It's a ring topology with three, sure. And then, I don't know, you could imagine like the top of the triangle. Maybe I'll connect to one more computer, but it's only connected to me. Now, it's like a triangle with a thing dangling off it, right? And so that is the shape of the network when someone says network topology. That's the shape of the network. We can kind of compare one ring technology with three nodes, another ring technology with topology with three nodes. They're kind of the same in the sense; so that sort of discrete version of topology. Now, there's a problem with that when you come to continuous space. Like, I don't know if you remember Anthony Aguirre, I think it was episode, was it 83, when I asked him about space? Because we still don't know whether…
Aaron: Yeah, 83 and 84.
Max: Yeah. So 83 was the physics one. And so I asked him whether like feel physical space really is real numbers, and I don't think it is. But I think like, yes, that is definitely the best way we have of measuring physical space. So in a network topology, I can ask you directly, “What are your neighbors?” Right? Let's say it's a computer network, right, and you’re on it...
Aaron: I know in that three-node ring topology that if I'm node A, node B and C are my neighbors.
Max: Exactly, exactly. So continuous space, you've got a problem. Because, let's suppose we're dealing with real numbers, okay? The real number line—all the numbers. What are the neighbors for the number 1? Okay, you could say, “Well…” you know, first guess might be, “Well, maybe 2 is neighbor to 1.” Well, no, there are lots of numbers between 1 and 2. There's 1.1, 1.2, 1.3.
Aaron: This sounds a little bit like Zeno's Paradox.
Max: Right.
Aaron: We can just infinitely subdivide the space between.
Max: Right, right. So you can't actually talk about what the neighbors of 1 are, and this is why the mathematics on top of that underlying topology seems complex at first, and they don't explain this, which to me is weird because it's actually a very interesting problem. And mathematicians, I think they kind of came up with the solution to this rather than like, you know, the 1800s, early 1900s, but the general solution is like in something called a neighborhood. So I can't tell you directly what the neighbors of the number 1 are, but I can tell you if I have a set that's got them all. So, let me give you an example. Like, let's suppose that I'm talking about a set of all the numbers between 0.9 and 1.1, okay? Include 0.9 and 1.1—let's say they're not included—but we're including all the numbers between.
Aaron: Okay.
Max: I kind of now have the point at 1 sort of suspended in jello there. I know it's in the interior of that set. So that set is therefore called a neighborhood of 1.
Aaron: To call it the confidence bounds is probably an inappropriate term, but that's kind of how I'm visualizing how wide that is is completely arbitrary.
Max: Well, no, no, no. So, you don't just have one neighborhood. Anytime you have a set that kind of surrounds the point—so it's not okay if the point’s on the boundary, it has to like surround the point.
Aaron: Right.
Max: It's like got all the neighbors. That's always called a neighborhood. And so neighborhoods have to have certain properties. Like if you take the intersection of two neighborhoods, it's still a neighborhood, for example.
Aaron: I assume that a neighborhood doesn't require the point you're concerned with being in the center of it.
Max: No.
Aaron: As long as it's not on the edge
Max: Exactly, exactly.
Aaron: Oh.
Max: Yeah.
Aaron: But then you don't even know how close you are to the edge; you just know that you're not on the edge.
Max: Exactly. Yeah.
Aaron: So it's bringing to mind the Heisenberg uncertainty principle, that I can know I'm inside of a group, but I can't know where I am inside the group.
Max: Yeah. I mean, I guess so. So it's more like, in a network topology, I'm going to give you a set of computers, and you look at all your neighbors, and if all your neighbors are in that set, then you're like, “Yep, that's a neighborhood.” And this is just the equivalent idea in continuous space, but whereas a network topology, I have to actually count the individual machines here—I can't do that. There is no individual machine. I just have to know whether I'm surrounded or not.
Aaron: So if the question is, I don't know what it is about the way you just explained it, but it invoked in my mind drawing curves and circles around things.
Max: Yeah.
Aaron: So does that bring us back into the shape side of this?
Max: Well, yeah. I mean, you could draw any shape behind it, but this is why continuity is such difficult concepts because you're not talking about going from this point to this point to this point to this point, but you can say things like, “Hey, if I have a neighborhood that's like small enough, then something's gonna happen. Like, I'm gonna get…” Sometimes, you can't talk about something happening at a point exactly, but in terms of like, you know, calculus, in terms of limits, you could say, “Hey, if I have a neighborhood of that point, if I'm like in all the points surrounding it, then I could talk about something happening.” And usually, you want those neighborhoods that you're—so a neighborhood could be any size. Like, you know, you could say, “Give me all the numbers.” Well, that's obviously a neighborhood, but usually, the neighborhoods you're interested in are the ones that are very small.
Aaron: I'm trying to think of a contextual example for using this, for why you would want to move from a discrete network topology approach to this continuous topological space approach, and I'm grasping at straws here.
Max: Yeah, there's a lot of—it's interesting—there are a lot of times in mathematics where we jump from discrete to continuous. And sometimes, you get interesting insights into continuous, and then when you get back to computation, you have to go back to discrete again. So wouldn't you admit that like, sometimes, when looking at the continuous analog of something, it's more you get insights that you wouldn't necessarily have otherwise? Does that kind of make sense?
Aaron: Sure, sure. I mean, even at the simplest level, it allows, you know, interpolation and extrapolation in a way that the discrete model doesn't really,
Max: Right, right. So let me, I'm trying to think of like...I don't know if I should even try to give the definition of continuity, but maybe I'll try, right. It's like, okay—now I talk about continuous math, which is, I don't even want to start talking about functions and stuff now—but it's like, okay, let's suppose I am measuring something on this space, okay. And at the value of 1, let's say the measurement or any point, let's say the value at some point A, the measurement is 6, okay? Then I should be able to, and let's say I want to be very close within 6—let's say I want to be within a millionth of it—I should be able to create some neighborhood around that point where it's always going to be within those bounds, if that makes sense. So it's like, you're not like jumping uncontrollably. In other words, you should be able to get close enough to the point where the values that are in the neighborhood should get close enough to 6. Am I completely losing everyone here?
Aaron: I think that makes sense.
Max: Okay, okay. I mean…
Aaron: I don't know if we're over or under explaining it here.
Max: Well, this stuff took me like years to understand what they're talking about. So if you even have an inkling of what I'm saying, then I'm happy.
Aaron: So, I don't know if we want to dig more on the continuity and topological space here or if we want to take the jump to the pointless topology.
Max: Right.
Aaron: So what on earth is that? And if this whole concept of topology, in particular, is coming out of at least in the discrete model, we start off with these points and the relationships to each other is pointless topology simply moving to the continuous model or…? Because it certainly sounds like an oxymoron.
Max: Right, right. So, no. Most topology is not pointless. Like you start with the points, and then you ask, “Okay, what are the neighborhoods of each point?” Pointless topology gets rid of the points entirely. Or it actually doesn't get rid of the points entirely, but it actually doesn't take points as the main structure that you're looking at. So first of all, there's this concept in topology called an open set, and an open set is, it's a neighborhood of all its points. So that basically means that like any point inside the open set is in the interior, and you could move, you can wiggle a little bit in every direction. So let's suppose—we're not talking about the basketball now—we're talking about the interior of the basketball, okay?
Aaron: Okay.
Max: So, every point in the interior, I could go a little bit in any direction and still be inside the basketball because I'm not including the boundary.
Aaron: That makes sense
Max: Okay. So, a lot of the results in topology can be, it kind of comes down to looking at what the open sets are in that system. And so, there's this one straight of thought, where we could say, “Okay, what if we just get rid of points entirely, and just say, “Okay, here are the open sets.”” So, for example, in terms of numbers, those open sets would be what are called open intervals. So, for example, it would just be like stuff that doesn't contain the boundary. So let's say all numbers between 0 and 1 but not including 0 and 1, that would be an open set. But if it didn't include 0 and 1, then it would not be an open set. And then, of course, like you could take unions of them, so you could have like a few of them, and they would still be open.
But essentially, the idea’s let's just consider open sets as the fundamental unit. And then yeah, you could use those to define points, whereas like it's a point is the collection of open sets that contain it, and then there are certain rules that that has to obey. But essentially, the idea is that an area of a certain extent has to be considered. So you can't consider points; you actually have to have a little substance in there, which, if you think about it is, actually, in some ways more like closer to describing the real world. Because any measurement that you take has some error bounds around it. And so you're never actually talking about points, you're usually talking about like a little bit of space.
Aaron: So the mental image that I have right now, and tell me if this is completely off base, but I'm picturing a Venn diagram.
Max: Sure.
Aaron: For our listeners, a Venn diagram, it's that classic you've got two circles, and they overlap a little bit in between, and it, you know, describes two different things and the features they have in common. So maybe round things and sports equipment and basketballs would be in the intersection there because they are both sports equipment and round, whereas baseball bats would not be in the overlap. So I'm envisioning that this…
Max: But they're all type topologically equivalent, but not...I’m now confusing things. But okay, go ahead.
Aaron: So I'm envisioning this pointless topology approach, if I want to describe the equivalent of a point there, is that I just need to keep drawing more and more circles on that Venn diagram so that I can narrow and narrow and further narrow the section right in the middle where all these circles overlap and try and more clearly define and constrain that ever-shrinking space in the center. Is there a kernel of accuracy to that?
Max: Yes, yes.
Aaron: Okay.
Max: So here's a mind-blowing thing, but it's—I don't even know if I should get into this because this is gonna be way too complicated—but okay, so with open sets, you have, you know set union and set intersection. I don't know if I have to define that here. But union is like, you combine the two sets together, so you get something bigger. And intersection is like, you know, in the Venn diagram, it's like the little piece in the middle. It's like what's the…
Aaron: The overlap.
Max: What’s the overlap, the exact…
Aaron: Or the commonality. Yeah.
Max: Right. And so, the set of open sets are, so a topological space of open sets is it forms something called a lattice—a bounded lattice in mathematics—which means you could take intersections of them, you could take unions of them, and then there's a top and a bottom, there's the whole thing, and then there's the empty set, right. That's the bounded part. And so if you map that one lattice on to another, you could say that it preserves union and intersection where it's like, okay, you take the union here, and then you map it to another lattice. So every point in one lattice is mapped to another point in another lattice. And like if the unions still work and intersections still work and all that after you've mapped it, then you say it's a lattice preserving map, right? This is all like category theory stuff.
So anyway, I don't know if anyone's following me, but I'm just gonna keep going here. So the Boolean set, which is there’s just a set of 1 and 0, is also a bounded lattice, right? Because you have 0 and you have 1, so you have empty and you have the whole thing, and it's also a lattice because you could take like the union of 0 and 1 is 1, and the union of 0 and 0 is 0. And so you could take unions and intersections there. So if you take the set of open sets, or if you take a lattice of open sets, and you do a lattice preserving map onto a Boolean space onto binary space, that map actually is a point.
Aaron: I feel like you lost me in the last step there.
Max: Right. So it actually defines, if you can define such a map, it actually ends up defining a point in the original space of open sets. God, topology. It’s so crazy. And this is like category theory now, so I don't even know. Yeah. This is like, and I'm not necessarily the one to talk about this either because there are definitely mathematicians who could have been in way deeper than I have done it like, you know, professionally, but that's just the way I see it.
Aaron: So let's assume for a minute that I've understood what you're laying down here.
Max: Yeah, I might have to draw it out.
Aaron: Let's take a step back. And you mentioned this is useful for machine learning.
Max: Well, I don't know if it is.
Aaron: Or for AI.
Max: Like no one uses it, so who knows?
Aaron: Okay. I guess. Well, let's take a further step back then.
Max: Okay.
Aaron: And for those of us who may not in our spare time want to geek out on infinitely cool mathematical ideas, what's the relevance of this to our average listener? Or is that a trap question?
Max: Well, I mean, it says it right in the name pointless topology. I don't know if it has a point. But, well, I guess topology, in general, has lots of points. But, you know, I just think it is an interesting way of looking at problems and looking at the world where you're just thinking about the connections between things, and you're kind of setting aside your notions of distance and form. And then in some ways, it's kind of a different way to approach problems, and maybe it's a way where you can kind of look for bias or eliminate bias.
Aaron: So when we say we're not concerned with distance, but we're concerned with like the shape...
Max: Well, not—I shouldn't even say the shape—I should say like the form. Because you could change the shape, you just can't…
Aaron: So, what is form in this context?
Max: You usually say, “You just can't cut and glue.” Or if you do cut and glue, you're changing the topological space.
Aaron: What I was going to try and reduce this to—we don't care about distance, it's about what else you're connected to. Is that?
Max: Exactly.
Aaron: Okay.
Max: So if you cut, you're removing connections. If you glue, you're adding connections.
Aaron: And so that knowledge of what those connections are, that defines, I guess, the possible shapes that could be taken on.
Max: Exactly. Yeah, yeah, yeah. So then you could take on, then you're like, “Okay, now we know the connections that would take the multiple shapes.” So you might not want to get into the mathematics, but it's an interesting idea to think about problems where it's like, “Okay, you know, the connections are all that matter so let me see if I can think about that in a different way, without putting aside my preconceived notion of which connections are more important and which aren't,” which is sort of a distance metric, right. Because if you sort of say—distance is sort of saying, you know, which connections are closer and which are further away.
Aaron: Okay, so the other thing that's popping into my brain right now is, and it's been awhile since I've looked at any of the white papers on this, but there's—I might have the terminology wrong—but some sort of network analysis in a communications paradigm, which has been used in intelligence work in the last 10, 15, 20 years to look at, you know, “You take person A, and who are they talking to? And then who are all those people talking to? And who are those people talking…” and you kind of move out in, maybe not concentric rings, but an expanding spider, and see what the connections between different players are. And that's one way you can determine what types of things someone might be involved in, even if you can't access the content of their communications but just by looking at who they're communicating with…
Max: Sure.
Aaron: That could potentially fit into this, that you're not necessarily concerned with, you know, how physically far apart they are if they're talking to each other—that kind of negates the disorder—but also you're not so concerned with how much they talk with each other, as you are that a connection between these two nodes exists.
Max: Yeah, yeah. Or maybe you do care how much they spoke to each other or how recently or how long the calls are, but you don't really know how to turn that into a score right away. Yeah, and so this is graph theory, this is like, you know, social network theory, and it's related—it is topology. It's highly related to topology, but you know, it's the discrete side of it. And so yeah, I think it's exactly the right place to—I think what popped into your minds is exactly the right way to be thinking because it's like we have the connections, but we don't know what they mean yet, and so let's see what the connections look like. And then on top of that, we can sort of give them weight after just looking at the connections themselves, if that makes sense.
Aaron: Okay. So I think maybe we've landed on…
Max: Let me give an example. Like, let me take this forward. I think this is important. Like, let's say someone says, “Okay, the number of times that two people speak to each other, that's how close the connection is.” And that's what I'm going to use, and there's totally stuck on that, and then they're not coming to the right conclusions. And then when someone says, “Well, wait a minute, this is a network. We don't really know what the distances are. Let's try to change around a little bit. Let's try to do different things.” Then maybe they can find B or whatever they're trying to find more easily. I don't know. Does that make sense?
Aaron: I think so.
Max: Okay.
Aaron: Yeah. This is a big thing to wrap our heads around.
Max: Yeah.
Aaron: Perhaps we will have to revisit it at a later date and see if we can put some more contextual spin on it. But yeah, this has been interesting.
Max: Wait ‘till we talk about knot theory.
Aaron: Actually, that was one of the things that popped in my head when we were talking about kind of contorting shapes.
Max: Yeah.
Aaron: Because that, you know, if you're taking your mug and flattening it and twisting it, and that very much gets into the realm of knot theory, which other than, like a 10-minute video that I've watched—I have no deep understanding of—but it looks super cool.
Max: Yeah, well, it is. I mean, so a knot is just if you take a string and you tie it, you glue the ends together.
Aaron: And in this context, you're talking like literally a string. We're not talking about like string theory in physics.
Max: No, no, no.
Aaron: Or strings of letters. You're talking like the concept of a physical piece of twine.
Max: Righ, right. Exactly. And so it's like, can you make them look similarly? Oh, here's, here's a mind-blowing thing about it. Think about a doughnut, okay, the surface of a doughnut, and, you know, imagine it's really long and skinny. So let's say it's like the crust of a pizza. Okay?
Aaron: Okay.
Max: Now, you gotta get that image in your mind. Okay. Let's suppose that I cut it at one in one point. So I cut the crust of the pizza..
Aaron: But just on one side of the ring.
Max: Just on one side of the ring, and then like I pull it out and make it a long line. Then let's suppose I tie a little knot, and I re-glue it together. Okay?
Aaron: Yeah.
Max: So now I have a knot. So, you could actually...now, remember the moving cities example I gave where we have the same neighbors as before.
Aaron: Yep.
Max: If we all lived on the surface of the old tourists, the old doughnut, the old pizza crust, and then we all moved into the tied up one that is topologically equivalent because we'll all have the same neighbors. But you can't continuously deform the first one into the second one without cutting and pasting.
Aaron: Right, as you say, that you can't get back to the way it was before.
Max: Right.
Aaron: Without those...yeah.
Max: And so those are, right, those are two different types of topological transformations. One's like, “Hey, can I make the whole jump and keep all the neighbors or do I have to like continuously deform it?” And those are—there are technical terms for this—but those are actually two different questions, which is really interesting.
Aaron: So this is absolutely fascinating, but it again brings me back to the question, okay, other than esoteric mathematics, what use is this?
Max: Yeah, I mean…
Aaron: I'm sure not theory. It must have some application to either something more fundamental in mathematics or some real-world application, but I can't mentally make the jump.
Max: Yeah, I don't. I'm not sure either. It's just interesting. I think it does have some applications in terms of string theory like in physics, but it’s speed—that's beyond me. I don't really know.
Aaron: Well, perhaps we need to find a knot theory expert to come talk some sense into it.
Max: Or a listener.
Aaron: I’m very interested to listen to that. It's a little bit different than our normal fare, but an episode that doesn't start off with talking about politics is always welcome.
Max: Well, yeah, so this is a very different episode. So how do you think it went?
Aaron: I think...I mean, I certainly understand topology and pointless topology better, but I'm still not sure. I like the idea of kind of holding this in reserve as a way to think differently.
Max: Yeah, I've been doing that for five years.
Aaron: But I'm not sure how to trigger that in the right moment.
Max: Alright. Well, we'll think about that. Alright, so I think that's all we wanted to say about this topic. We have anything else for today?
Aaron: I'm gonna go put my brain on ice after this.
Max: Good luck. Well, you know, we're working out the brain. We're getting out those local maximums.
Aaron: Yeah.
Max: Or reaching different local maximums, you know.
Aaron: If it's not confusing, you're not learning.
Max: Exactly, exactly. Alright, maybe I'll post some of the books that I've read on the show notes page, localmaxradio.com/133, we're at now.
Aaron: Maybe if you can scrounge up some diagrams that aren't too obtuse, that might help.
Max: Yeah.
Aaron: For people trying to follow along.
Max: I can't, I hate making diagrams for the website. I know I should, but it's like it's too time-consuming. You'll have to find somewhere else, but I'll link to some books. Oh, I did want to talk about last week's episode. I know you listened to that with Rob Bernstein, and this is a very different episode today, but that was a lot of fun. I hope people enjoy—I would like to see the intersection of people who enjoyed today's episode and last week's episode because that would be a great local maximum.
Aaron: Those are the kind of people you want to invite over to your cocktail party or barbecue.
Max: Yeah.
Aaron: They'll have interesting things to say about pretty much anything.
Max: Back when we used to have events and stuff like that, but in 2021.
Aaron: You know, I get to do it, but I've had a couple of people suggest, “Oh, maybe we should do like a socially distant thing,” you know.
Max: Oh, I don't.
Aaron: At your house or, you know, in a park.
Max: I don't do socially distant anymore. That is so May 2020.
Aaron: Well, so I recently…
Max: I've survived this long. What's it gonna take? We've got less than like .3% positive rate in New York City now, and the testing, it's like how low do you need it to get?
Aaron: Oh, yeah.
Max: And it's not like it's growing exponentially anymore.
Aaron: Not to get into topics that will not be evergreen, but I...
Max: Well, whatever we're done with the topology.
Aaron: Yeah, I hope that the rest of the country can be as confident in where they're sitting as you are in the near future.
Max: Yeah.
Aaron: But so what are you saying, speaking of surviving, I recently survived another year, and we briefly contemplate.
Max: Alright, happy birthday.
Aaron: Do we want to invite people who aren't that far away to come over and like, “We'll leave some cake out on the deck, and you can come take a piece and then,” you know, “step back 20 feet and stand out in the deck, and we've got some chairs,” and it's, you know, it was just too much hassle, especially with the little kids and everything going on. And we wimped out and didn't do it, but my work is talking about we're sick of doing like virtual happy hours that we need to do.
Max: We’re tired of it.
Aaron: We need to like brown bag it in a park somewhere or something.
Max: Yeah.
Aaron: So we'll see when I break the seal, with kids going back to daycare next week. It's only a matter of time.
Max: I mean, I feel like the number of people who have it will go to zero, and people still be all crazy.
Aaron: Well, yeah. I mean, nothing in this situation has convinced me that we're going to solve crazy.
Max: I mean, if it gets worse, it could get better, right? No? Or does it only get worse?
Aaron: There's, you know, the only way out is through philosophy, but I don't know if I want to embrace that for everything.
Max: I mean, I think that...well, you know, it's weird. I go to the Foursquare office these days. It's like a 150-person office in Flatiron in Midtown Manhattan, and I'm the only one there. Now because I've moved close by so I go, but it's like, it’s literally I have the whole office to myself. So I guess I don't have to wear a mask if there's no one there, but I love how like when they invite us back, “Remember to,” you know, “stay two desks apart, and this and this,” and like my desk is one of the ones be X’s on it, like don't use that desk. It doesn't matter, I just sit down at my desk and use it because there's no one else there in the whole place.
Aaron: Oh, you rebel.
Max: There might be one other person, but...
Aaron: Flaunting the signage?
Max: Well, I'm using the best monitor there, and I'm using somebody else's iPhone charger, and why not? I'll charge my AirPods too because there's another charger like another three desks down. Nobody's there. It's great. I mean, I wish some of the people I was working with were there because that I would get more done, but it’s so hard to focus alone. I think, I don't know. I don't know what life's gonna be like, but I think remote is here to stay but not like this. Not like this.
Aaron: Yeah. I don't know that many people in kind of the business manager role, but I did recently talk with a friend of mine who owns his own business. And he said that, I think it was just this past month, they got out of their lease, and they're not going to have an office for the remainder of 2020.
Max: Wow.
Aaron: And then, you know, I think the catalyst for them to, again, procure office space is going to have to be pretty high. It's definitely making changes.
Max: I feel like it's so hard for me to get like started on a project or a task when I'm sitting at home. I have to really mentally like, you know, have tons of willpower to do it. And I don't know, maybe that's just me. I mean, I guess the podcasts, I found podcasting is the one thing that I'm able to focus on
Aaron: At least, usually, it doesn't feel like work. It's a little different.
Max: Right, right,
Aaron: And while we do have to provide our contractually-obligated, you know, once a week, somewhere between 20 minutes, and well, we're creeping up on an hour here. But, you know, we're not filling out a timecard for this. So it's a little bit different.
Max: Exactly, exactly. Alright. Cool. I think that's all I have for this week. So unless there's anything else, I think it's time to head out.
Aaron: Yeah, we'll see what next week has in store for us. We'll be out the other side of the DNC convention and be thrown right into the madness that I'm sure will be the RNC convention.
Max: Yeah, back to, back to that. Well, it is the season, the one of the four-year season. Yeah, I should get...
Aaron: The 18-month long season.
Max: Yeah, I should get...yeah, but this is the real deal now. I'm gonna try to find some guests too. I don't have any specific guests lined up, but I have some people I can ask, and I'm excited about it, so we'll see. We'll see who I can get over the next few weeks, but, you know, maybe I can get a topologist or a category theorist on the show. That would be great.
Aaron: That would be pretty neat. I imagine we could also have some pretty interesting conversations about polling as we enter the end times.
Max: Oh, yeah.
Aaron: So I don't know if there are any listeners out there who are pollsters? Nate Silver, I know you're listening. Get in touch with us. We have questions,
Max: Maybe I’ll get Alex Andorra back on the show to talk.
Aaron: Yeah, that's right. Well, and the other thing I forgot to mention is with the convention this week, a lot of Metaculus predictions I’ve made have been resolving.
Max: Oh, we’ve got to talk about that.
Aaron: So it’s time to go and make some new ones. Oh, that's right.
Max: We got to talk about that because—so I made one. I had, first of all, I didn't realize…
Aaron: You drafted a question.
Max: I drafted a question. I also had one that got resolved yesterday, which I didn't realize was still open, which was, “Will Joe Biden be the Democratic nominee?” And apparently, I said yes, but I said yes a long time ago when it wasn't over, and it was finally decided today, and I got a lot of points for it.
Aaron: Yeah, most of the ones that have been resolved in the last week or two were ones I've completely forgot I made that prediction because it was so long ago.
Max: Yeah. And so I made a question asking whether there will be a Trump-Biden debate. Because I saw a lot of people saying, “Oh, it might not happen,” or “It shouldn't happen,” or “Biden shouldn't debate Trump,” and whatnot. And so I thought it'd be interesting to put that out to that community, and I think it's currently at 80%. My prediction is higher. My prediction’s like 92%. I think it's almost certainly going to happen, but I could be wrong.
Aaron: Yeah. Well, let me take a peek here. So I need to find it here. Okay, here we go. It's sitting, it looks like the community prediction is that 85% right now. And we've got, 44 predictions have been made. So yeah, if you're hearing this, go out there and make your prediction. You've got ‘till September 15 to lock that in.
Max: Yeah. Oh, no, it's an 85. Okay, I put mine at 90, I guess. I think I was thinking 92, and then I saw the community at 80, and I was like, “I don't know, that's one or two points.”
Aaron: My current prediction is also at 90.
Max: Oh, okay.
Aaron: But I may revise it in the next couple of weeks.
Max: Sure. I think…
Aaron: We've got just under a month.
Max: I lost money in predicting because I didn't think Kamala Harris would be the Vice President nominee.
Aaron: I've heard that the VP nomination market was insane right up until the last minute.
Max: Yeah, I wasn't even paying attention to the last minute, but I really thought that that wasn't—well, I don't know. I'll try to think about where I went wrong there.
Aaron: Well, we gotta, yeah, identify the flaws in our methodology and see if we can improve our predictive process the next time around.
Max: My methodology for that was just to look at it and be like, “Ah, they can't do that.” But I guess I was wrong. They wouldn't do that. Oh, well. That's where you get into trouble and predict it.
Aaron: It's interesting, the differences between predictions and those on Metaculus since there's no money on the line in Metaculus, but it feels much more scientific.
Max: Yes, I think predicted...first of all, it's not a real market because there are such caps. So you don't have really smart, big money players in there. And I mean, some people might be smart but also like the comments, it’s all just political like trash talk.
Aaron: You get some of that on certain questions in Metaculus, but I feel like it's moderated away from that for the most part.
Max: Yeah. Yeah, I feel like Metaculus is being more scientific.
Aaron: This is a topic that I think we will be able to discuss at least several more opportunities in the next couple of months. So we shouldn't just invest too much in it just yet.
Max: Alright, alright. Cool. Well, yeah, this would be a good time to solicit input because we went over a lot on the show—localmaxradio@gmail.com, if you want to weigh in. Alright, have a great week, everyone.
That's the show. Remember to check out the website at localmaxradio.com. If you want to contact me, the host, or ask a question that I can answer on the show, send an email to localmaxradio@gmail.com. The show is available on iTunes, SoundCloud, Stitcher and more. If you want to keep up, remember to subscribe to the local maximum on one of these platforms and to follow my Twitter account @maxsklar. Have a great week.
