Trying to Understand the Second Paragraph of “A Concise Course in Algebraic Topology”

In my attempt to understand the first paragraph and the notion of a topology (and a topological space) I realized that there was something I wasn’t getting. I understood the definition of “open” subsets of X, I also understood the meaning of “neighborhoods,” and I felt like I was getting a grasp on the axioms.

But what does it mean to have a “notion of nearness”?

Now, I think I’m starting to get it. In Calculus or Real Analysis you define a continuous function in terms of distance and the “I can get closer” game. But the point is that you don’t need distance in order to define continuity or a lot of those other analytic notions. All you need is a “notion of nearness,” and if open sets and neighborhoods make sense in your space, then you can usefully obtain that notion of nearness.

The usual Calculus definition of a continuous mapping goes like this: if we want all the values in the range to be within a certain neighborhood — no matter how narrow the neighborhood — we can always find a suitable neighborhood in the domain whose points all map to that band of points in the range. In this, a neighborhood is defined in terms of distance — wanting to be in a certain neighborhood in the range is being a certain (potentially tiny, epsilon) distance away from a point in the range.

But who needs to define a neighborhood in this way? We can replace this metric notion of neighborhood with a topological one, the one that we struggled to grasp in the previous post, that really comes down to open subsets. So, in short, a function is continuous at a point in a topological space if, no matter the neighborhood in the range (call it Y), there’s always a neighborhood in the domain (X) that the function maps to a neighborhood that’s a subset of the original (Y).

I should probably be trying to make these images myself rather than stealing them from wikipedia but whatever:

So now I’m thinking of topological spaces as metric spaces, minus distance. Which explains something I never really understood, which is why in topology you can deform all sorts of things into other things. Why should a coffee mug and a doughnut be equivalent in a topological sense? The answer is that distance doesn’t exist in a topological space. All we have is open sets and subsets of open sets which stands in for our notions of nearness. As long as a mapping (like a homeomorphism) preserves the open set structure — the idea that things that start out near each other should remain near each other, relatively speaking — then there is a topological sameness that is preserved.

Honestly I should probably just be learning topology instead of trying to make sense of this algebraic topology text, but let’s see how far we can go. The next paragraph makes the jump to algebraic topology.

Trying to Understand the First Paragraph of “A Concise Course in Algebraic Topology”

I.

Wikipedia is here, and quite helpful in expanding on this definition. Here is my attempt to rewrite it in my own words, fleshing it out in ways that are helpful and natural to me.

What is a topological space?

Imagine that you have some collection of points that are in the Cartesian plane. Maybe it’s a circle that’s centered around the origin. Maybe it’s a line, like y = 2x + 3. Maybe it’s a grid of points falling perfectly along the chassis of the plane.

In a way, we can also imagine the city that you live in or the state or country as a collection of points. My apartment is one of these points in space, and I’m part of a neighborhood — Washington Heights. But there are other ways of talking about where I am. I’m not just in Washington Heights, but in NYC, in NYS, in USA, on Earth. There are points, arguably, south of where I live that are a part of two neighborhoods — Washington Heights and Harlem. These neighborhoods aren’t coextensive, but they do have some overlap.

Maybe there are some folks living out in the sticks that are part of no neighborhood. They’re off the grid, so to speak.

Let’s imagine a computer program that knows how to assign every point in a map of NYC to the neighborhoods to which it belongs. Let’s call that program N. You give N a point, and it gives you a list of neighborhoods.

Imagine if we let N loose on any of those spaces that we were talking about before — the disk, the grid of integer points, the portion of a line — and then let’s say that any of those spaces is a topological space if when N is applied to the space, nothing weird happens. Here is a checklist of four weird things that could happen:

• You apply the computer program N, but there’s a place that gets assigned neighborhoods that it’s not in. Like, maybe the program assigns a spot in the middle of the Hudson River to Washington Heights, even though a random spot in the river isn’t really a part of our neighborhood.
• Imagine that N says that my apartment is in Washington Heights, but not in NYC. That’s messed up. Bad N. If you can imagine a place like that, it’s not a topological space.
• If I lived a bit further north, I might be a part of two neighborhoods: Washington Heights and Inwood. N should assign me not just to Washington Heights and to Inwood, but to a new, hybrid neighborhood: Inwoodton Heights. If N can’t really invent such a neighborhood, then I don’t really live in a topological space. This one starts moving away from my regular intuitions about neighborhoods, clearly…
• …and so does this last test. As I keep on mentioning, my apartment is in a neighborhood Washington Heights. If N assigns my apartment to WaHi, then it should also report that there’s another neighborhood that I’m a part of that’s more local than WaHi, like my block. My block has the property that (a) I’m in it, it’s my neighborhood and also (b) everywhere on my block is also part of Washington Heights. If there isn’t such a place, I don’t live in a topological space.

Moving away from apartments and cities, let’s think about the x-axis. Real estate agents who are trying to get people with $to move to a place decide what neighborhoods mean in the real world. But what does a neighborhood mean on the x-axis? Let’s define a ‘hood of a point (like 3.1) to be any open interval (like between 3 and 5) that includes the point. Let’s make sure that our computer program N would work alright in such a place: • If you’re a neighborhood of a point, that point is in it. Yay! • If you’re in an interval (like between 3 and 5) and that interval is in a bigger interval (like between 1 and 6) then you’re also in the bigger neighborhood. • If you’re part of two neighborhoods (like between 3 and 5 and also between 3.05 and 7) then you’re also part of the overlapping neighborhood (between 3.05 and 5). • If you’re part of a neighborhood (like between 3 and 5) there is a smaller, more local neighborhood that you’re part of (like between 3.05 and 4.95) and, more to the point, there always will be. Huzzah — it’s a topological space! II. One more twist: this way of thinking about topological spaces is not standard any longer. People prefer to talk not about neighborhoods but instead about open sets. So we have to make sense of that, even though it’s been nearly a decade since we took Real Analysis. Another wise wikipedia page says, “an open set is an abstract concept generalizing the idea of an open interval in the real line,” which makes sense. It tells you the delimit of a set, but it doesn’t include its boundary. This isn’t enough, though. Since we defined a topological space in terms of neighborhoods, we want a definition of “open set” that relates “open set” to neighborhood. We need to connect these two concepts. [Here’s where I got a little bit lost, so I went looking for help from another source. Google google, ended up at the Math Stackexchange.] It doesn’t take much to merge the two concepts, as far as I can tell from what I’m reading. To fit the rules of neighborhood assignment, your potential neighborhood has to pass the following test: if a point is assigned a particular ‘hood, the point actually has to be in that ‘hood. (Fancy talk: if N is in N(x), then x has to be in N.) An open set adds one slight additional requirement: if you’re in N, then N’s your neighborhood. All this eliminates is the possibility that a neighborhood is “too big,” including not just Washington Heights but also a random stretch of the Hudson River where nobody lives. This is a definition that will do the trick for us, and allow us to connect the old-fashioned (but apparently useful for beginners like myself) definition of topological space to the new and trickier (but apparently useful for topology pros) definition that is couched in terms of open sets. Here’s the open set version: still imagine a map of some space (like NYC) and still imagine that the map contains a bunch of points (like my apartment and other peoples’ apartments) and still imagine that there are ways of grouping those points that come with the map (like neighborhoods)… …or don’t, and instead imagine some section of the x-axis and the Cartesian plane, and imagine that there are ways of making subsets of that section of the line or plane… And now there are three rules about the neighborhoods (or subsets): • The empty set is in it — i.e. the neighborhood of nobody is a neighborhood • You can’t escape the space via the union of subsets — i.e. mush together any of the neighborhoods (subsets) and you’re still in a neighborhood (subset) • You can’t escape the space via the finite intersection of subsets — i.e. be in as many of the neighborhoods (subsets) as you wish to at once and you’re still in a neighborhood (subset) Not entirely obvious to me yet why these two definitions of topological space are equivalent. I see that they both have the intersection, I see how the intersection of disjoint sets would imply that the empty set is in the topology (but what’s wrong if there’s only one set in the topology according to the neighborhood definition?), and I guess the superset/subset axioms must cover the “closed under union” axiom of the open set definition? I’m going to let that slide for now. III. What are some examples and non-examples of a topological space? We already mentioned the x-axis, equipped with open intervals. In fact, for things like the x-axis — metric spaces — the textbook page already tells us that we can think of topological spaces as things where the neighborhoods are like open intervals. They’re open disks, or open spheres, that act precisely in the way you’d expect them to. No funny business, no weird stuff. As the textbook also says, we should think of this all as an attempt to point at what it takes to have a space that captures our feelings about “nearness.” So if something’s near, there’s always something nearer, that seems to be the most important part to me. But whenever there’s a new idea, we need some examples and contrasting non-examples to set our heads on straight, and this is no exception. The x-axis — i.e. the real line — is an infinite collection of points, but there’s no reason why a topological space needs to be infinite. And if our map of points is finite, there’s no reason why our computer program — our function — N has to be anything but a list of neighborhoods. So, suppose you have four points in your space: 1, 2, 3 . What neighborhood assignments would result in this being a topological space, and which wouldn’t? Here is a helpful image from wikipedia: • If you care your space into two sets — {} and {1, 2, 3} — then you haven’t really done much carving, but that’s a topological space, the trivial topology. That sort of captures the idea of nearness in an absolute way — everything is near, nothing is not, like a terrible party. (No idea if that analogy makes mathematical sense, by the way.) Thinking about the function N and the first set of axioms: the same and only neighborhood has been assigned to the three points and they’re part of it, there are no supersets or subsets, so trivially we’re done. • In contrast, if you carve your space into {}, {1, 2, 3}, {2} and {3} (bottom left) that’s not a topological space. From the point of view of the second definition, the issue is that the union of {2} and {3} isn’t included, so it’s not a topology. From the point of view of the first, the problem is that {2, 3} is a superset of {2}, and every subset containing a neighborhood must itself be a neighborhood. (Is this right?? It doesn’t feel right.) • What if your topology contains {}, {1}, {2}, {1, 2}, {1, 2, 3} (middle left)? The second definition is indeed sort of easier to use for these discrete examples. It’s pretty clear: the union of {1}, {2} is {1, 2} and that’s in. I need more examples, and wikipedia has a few more: • Let the space be all the integers, and let the collection of subsets be all the finite subsets of the integers, any list of them, plus the set of all the integers. So, for example, if you have a given point like -5, there would be all these open intervals that we’d want to say are the neighborhoods of -5, like “integers between -10 and 2” and “integers between 0 and -10000,” and we’d also say “all the integers.” That wouldn’t work, though, because you the union of a bunch of these subsets isn’t necessarily part of the topology. Take the union of all the finite sets that don’t contain zero — {-2, -1, 1, 2}, {1, 5, 6}, etc. — that’s an infinite set, but there’s only one infinite set in the collection (all the integers) and this can’t be that infinite set. Therefore, it can’t be in the collection, and this can’t be a topology. The tricky thing for me is relating this all to the talk that a topological space preserves our notion of “nearness.” Is it possible to relate each of these axiom failures to a failure of our notion of nearness? Personally speaking, my notion of nearness no longer operates when our space has just four points. Is there a good way to think about this that I’m missing? TO BE CONTINUED, WHEN I TRY TO READ THE SECOND AND THIRD PARAGRAPHS OF THIS TEXT AS I ATTEMPT TO UNDERSTAND A PROOF OF BROUWER’S FIXED POINT THEOREM. STAY TUNED In Love with Being Lost Sunil Singh writes: What is mathematics? If I would have answered this question before I began my teaching career, I would have been provided some cliched, stock answer revolving around its importance to many disciplines like science, engineering, economics, etc. — basically it being some practical workhorse. Now? Well, since I wrote a book about it, it is simply about happiness. Finding sporadic moments of bliss learning something new and wonderful about mathematics. Just playing with numbers, puzzles, brain teasers, conundrums, algebraic ideas, etc — mucking around in the sandbox of math. The more I know, the less I know. The less I know, paradoxically, the better I feel about my ideas about mathematics. There’s no doubt that mathematics involves happy moments. But does Andrew Wiles’ description of mathematics sound like happiness? Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. You go into the first room and it’s dark, completely dark. You stumble around, bumping into the furniture. Gradually, you learn where each piece of furniture is. And finally, after six months or so, you find the light switch and turn it on. Suddenly, it’s all illuminated and you can see exactly where you were. Then you enter the next dark room… Not that this is necessarily inconsistent with joy. Maybe you get your kicks wandering around anonymous dark mansions — there’s something for everybody. But you’ve got to admit, it sure doesn’t scream out FUN. So which is it: is mathematics all about happiness or about stubborn frustration? Well, why not both? If, as Singh suggests, we’d best think of mathematics as an “art” then we get to ask, what other art is only about happiness? I think educators sometimes emphasize feelings of joy and experiences of beauty in math at the expense of all the other things you can feel or see. For a while, I’ve been interested in visualizations of ugliness in math — is such a thing possible? And, as the Wiles quote shows, mathematics often involves far more than happiness. “Of course math involves more than happiness,” you say, “but the point is that happiness is the goal, the purpose, the carrot that mathematicians are always chasing.” Here’s an idea though: what if what we’re really in love with is the feeling of being lost in a dark mansion? Generalizing Circles and Ellipses [Reading and thinking about: n-Ellipses and the Minimum Distance Sum Problem, a paper that I was able to stick pretty well with until about the last third.] 1. Generalizing Circle Definitions There often comes a point in my geometry classes where I ask students to come up with a definition that describes a circle. It’s always interesting to see the ideas they land on. Some students try to capture the perfect curviness of a circle; others talk about the circle’s symmetry. When I share the “textbook” definition (which kids rarely come up with) there are sometimes sighs of relief or yelps of excitement in the room: Circle: Take all the points that are a certain distance away from a center point — that’s a circle. It’s a beautifully simple definition, and it’s one that leaves plenty of room for generalization. If we’re looking to generalize the circle, a great next step is to analogously define an ellipse. I like to imagine the center of a circle splitting in half and turning into the two foci of the ellipse. (Sorry for the glitchy gif.) We can define an ellipse in a way that is analogous to the “textbook” definition of the circle: Ellipse: Take all the set of points that are a certain distance away from two focal points — that’s an ellipse. Just to clarify, what I mean by “certain distance away” is we’re looking for the sum of the distances to the two focal points. So if our “certain distance” is 10, then a point that is 3 away from one focal point and 7 away from the other would be part of the ellipse. 2. Generalizing Circle Constructions Sometimes you see geometry problems about goats on a rope. The point being that the rope-stuck goat is restricted by a circle. That goat could trace out a circle, and you can also trace a circle with a pin, string and paper. You can also make a lovely ellipse with two focal points with some string and some pins: Why stop here, though? The definition of ellipse (let’s call it a 2-ellipse) can be expanded to include 3 focal points, 4 focal points, 176 focal points, really focal points, no matter what integer is. n-ellipse: Take all the points whose distances to each of n focal points sum to a certain number. This is the n-ellipse. James Clerk Maxwell was sixteen years old when he invented a contraption that generalized the pin and string method for constructing an ellipse. He figured out a clever way to wrap the string around so that you could use pins and string to produce a 3-ellipse. 3. Generalizing Equations The equations generalize nicely too. Circle (a.k.a. 1-Ellipse): $\sqrt{(x-a)^2+(y-b)^2} = D$ 2-ellipse: $\sqrt{(x-a_0)^2+(y-b_0)^2} + \sqrt{(x-a_1)^2+(y-b_1)^2} = D$ 3-ellipse: $\sqrt{(x-a_0)^2+(y-b_0)^2} + \sqrt{(x-a_1)^2+(y-b_1)^2} + \sqrt{(x-a_2)^2+(y-b_2)^2} = D$ n-ellipse: $\sum_{i=0}^{n}\sqrt{(x-a_i)^2+(y-b_i)^2} = D$ (Most students learn a different equation for an ellipse, defined in terms of minor and major axes. It’s not exactly a load of fun to analytically derive that from this equation for the 2-ellipse, I’ve found.) 4. Smallest Ellipses? Each of the definitions above leave room for a “certain distance.” But are there any restrictions on what that distance can be? For a circle, that “certain distance” is the radius, and the radius can be whatever magnitude that you like. Given a center, make your radius huge, make it tiny: there’s always a circle of those points just that distance away. For a 2-ellipse, though, this isn’t so. Imagine one focal point that’s one million miles away from the other, and imagine that we’re interested in the points whose sum of distances from the focal lengths is one inch. That’s simply impossible to achieve. Being one inch away from one focal point would necessarily involve being nearly a million miles away from the other. At some point, an ellipse might be too small for its focal points. So: what is that least distance, given two focal points? What is the least amount of distance that you can sum to, given focal points? What are these critical points of the distance sum function? For a 2-ellipse, the minimum distance should just be the distance between the two focal points. But what about for a 3-ellipse? or a 4-ellipse? an n-ellipse? I’m still having trouble with the analytic solution in that paper, but graphically you can just graph a bunch of different distances and try to see where things are headed. Here is a snapshot from this Desmos graph, which graphs a collection of distances for three foci (which you can move around the graph — try it, it’s fun!). This is a contour plot of the distance-sum function. We could also visualize this as slices of a 3D graph, where the “z” axis is the varying distance-sum. Here is a 3D graph of a 2-ellipse, showing how it bottoms out at a very narrow ellipse (practically a line segment) at a height of 2 — the distance between the two foci: And here is a 3D graph of a 3-ellipse, which also bottoms out at a certain distance-sum above 0: In contrast, here’s the 1-ellipse, a circle. It has no minimum radius: 5. A cool problem Say that you have three cities that are special to you, for whatever reason. Maybe you grew up in one of them, went to college in another, and have some family in the third. If your only priority was to be as close as possible to each city, where should you live? This is really, really similar to the question of where the minimum distance-sum is for an n-ellipse. (It makes sense that it starts getting circle-ish as we get farther away from the focal points. From a million miles up in the air, those focal points might as well be on top of each other.) Go to this graph and play around with your cities. You can also add more cities. 7. Now what? I’m not sure. I tried to write this a few different ways, but in the end all I could figure out was the info dump you see above. Is there a way to rewrite this so that there’s a stronger narrative? Should I come up with problems for each section to give readers a chance to stop and think about stuff? I think this needs to go on the back-burner, and I just need to write more about math and hope that eventually I figure a few more things out. Mathematical and Scientific Crises [This post is wildly speculative. File under ‘playing with ideas.’] On the continuing and fascinating saga of the ABC conjecture and Mochizuki’s befuddling proof: Five years ago, Cathy O’Neil laid out a perfectly cogent case for why the (at that point recent) claims by Shinichi Mochizuki should not (yet) be regarded as constituting a proof of the ABC conjecture. I have nothing further to add on the sociological aspects of mathematics discussed in that post, but I just wanted to report on how the situation looks to professional number theorists today. The answer? It is a complete disaster. From Andrew Gelman’s post on the ongoing replication crisis facing psychology and many, many other fields: At this point, certain earlier work was seen to fit into this larger pattern, that certain methodological flaws in standard statistical practice were not merely isolated mistakes or even patterns of mistakes, but that they could be doing serious damage to the scientific process. Some relevant documents here are John Ioannidis’s 2005 paper, “Why most published research findings are false,” and Nicholas Christakis’s and James Fowler’s paper from 2007 claiming that obesity is contagious. Ioannidis’s paper is now a classic, but when it came out I don’t think most of us thought through its larger implications; the paper by Christakis and Fowler is no longer being taken seriously but back in the day it was a big deal. My point is, these events from 2005 and 1007 fit into our storyline but were not fully recognized as such at the time. It was Bem, perhaps, who kicked us all into the realization that bad work could be the rule, not the exception. Two entirely different scientific controversies occurring in entirely different fields and of an entirely different nature. They have nothing in common. Right? Right? I don’t know if there’s anything here, especially as a non-expert in this (and any) field. (Sigh. I do love teaching, but sometimes I wish I had a knowledge base and expertise that I could really rest on. All this stuff of controversy makes me wish that I could really speak with authority on any of this fascinating stuff. A different lament for a different time…) OK, so let’s toss one more scientific controversy into the mix: Yann LeCun’s post about AlphaZero-style machine learning, and the opaqueness of AI’s solutions to human observers: The main message was, in essence, that the current practice in machine learning is akin to “alchemy” (his word). It’s insulting, yes. But never mind that: It’s wrong! Ali complained about the lack of (theoretical) understanding of many methods that are currently used in ML, particularly in deep learning. Understanding (theoretical or otherwise) is a good thing. It’s the very purpose of many of us in the NIPS community. But another important goal is inventing new methods, new techniques, and yes, new tricks. In the history of science and technology, the engineering artifacts have almost always preceded the theoretical understanding: the lens and the telescope preceded optics theory, the steam engine preceded thermodynamics, the airplane preceded flight aerodynamics, radio and data communication preceded information theory, the computer preceded computer science. Why? Because theorists will spontaneously study “simple” phenomena, and will not be enticed to study a complex one until there a practical importance to it. And maybe this is the tension that unites all three of these scientific controversies. We aren’t living in an moment marked by intense industrial or even informational explosion. Rather we are living in an age of immense theoretical constructions that outstrip any single person’s ability to understand them. On the corporate side this includes the algorithms of Google or Facebook. The controversy over machine learning is of the value of algorithms whose methods are unknown to the operators — e.g. facial recognition software whose methods for identifying faces would itself need to be studied in order to be comprehended. Now, the crisis facing psychology and other sciences isn’t exactly new and its causes include non-open data practices as well as publication bias and a commitment to conceptual replication. (More here.) But it seems that a lot of scientists were treating statistics itself like a black box. You collect some data, and then toss it into the statistics pot and, boom, out comes an effect. Science was using statistics as a big, mysterious theoretical edifice, and now we’re grappling with the aftermath. And maybe this is a way to think about the Mochizuki controversy. True, the edifice he’s created is impossible to penetrate and basically no one understands it. But that puts it in good company, along with machine learning engines and the statistical packages that psychologists run their data through. Maybe there was a time when science was about understanding, but now we’re trying to take science beyond mere understanding, and struggling to find the rules of engagement. My set theory book is getting kind of fascist Definition: A set is said to be pure if no individuals belong to its transitive closure. Well there are always individuals, right? Consider now the following axiom candidate. Axiom of purity: Every set is pure Fully pure? The main reason for the [axiom of purity] is that, as was discovered fairly early, it is is not necessary to assume the existence of individuals in order that set theory should act as a foundation of mathematics, while if we rule them out from the outset, we can simplify the theory. A TOTAL BAN ON INDIVIDUALS After 1963, not even set theorists had any use for individuals. Worse, there are proofs in set theory that do not work if we have to allow for them. So it is unsurprising that in the last 40 years individual have largely disappeared from view. FULL PURITY FOR MATHEMATICS. NO INDIVIDUALS. ALL ARE SETS. THIS WILL ENSURE THAT EVERYTHING IS WELL-FOUNDED AND REGULATED. However, we shall not follow this trend here. …ooh? The reason is that to do so would cut our theory off from at least one of its intended applications. It is by no means obvious what justifies the applicability of mathematics in general to what lies outside it…the most natural, if not only way, way to ensure that that calculus is available to be applied to counting non-mathematical things — chairs, electrons, thoughts, angels — is to allow such things into the theory as individuals. Phew. From Set Theory and Its Philosophy. The piranha problem A fundamental tenet of social psychology, behavioral economics, at least how it is presented in the news media, and taught and practiced in many business schools, is that small “nudges,” often the sorts of things that we might not think would affect us at all, can have big effects on behavior. Thus the claims that elections are decided by college football games and shark attacks, or that the subliminal flash of a smiley face can cause huge changes in attitudes toward immigration, or that single women were 20% more likely to vote for Barack Obama, or three times more likely to wear red clothing, during certain times of the month, or that standing in a certain position for two minutes can increase your power, or that being subliminally primed with certain words can make you walk faster or slower, etc. The model of the world underlying these claims is not just the “butterfly effect” that small changes can have big effects; rather, it’s that small changes can have big and predictable effects. It’s what I sometimes call the “button-pushing” model of social science, the idea that if you do X, you can expect to see Y. Indeed, we sometimes see the attitude that the treatment should work every time, so much so that any variation is explained away with its own story. In response to this attitude, I sometimes present the “piranha argument,” which goes as follows: There can be some large and predictable effects on behavior, but not a lot, because, if there were, then these different effects would interfere with each other, and as a result it would be hard to see any consistent effects of anything in observational data. The analogy is to a fish tank full of piranhas: it won’t take long before they eat each other. From Andrew Gelman’s blog. Read the whole thing, which I think has applications for education and education research. I sometimes think that there are two fundamentally opposed world-views that are currently competing for dominance: • The most important factors that impact culture, education, behavior are incredibly un-obvious and invisible. They are subliminal, subconscious, invisible, linguistic. Therefore, to improve the world, sweat the details. • The most important factors are big, obvious things. They are economic, material, visible, sitting right in front of our noses. To improve the world, improve these big things. I don’t think these categories break down cleanly along any of the left/right lines, but I do think the more educated you are the more you’re (we’re) asked to notice the little things. Maybe there’s a correspondence there? When I say people, I mean mathematicians On the wave equation and the beginnings of Fourier analysis: “Very soon after that, people began…when I say people, I mean mathematicians.” I only know Rudin from his analysis books, but come on, man! What, are you worried that someone will think mere people would be interested in this? Another not great moment comes a few minutes later: “D’Almbert raised a rather strange objection to that. This is a curious objection to my mind, because these were sophisticated people, and why d’Almbert couldn’t see…is hard to believe.” It’s a lecture on history of mathematics. Rudin warns us at the start that mathematical history is understudied, and that mathematicians aren’t exactly qualified for it. I’m glad he had that awareness, because history starts right here, at that moment of disbelief. The historian has the tools and penchant to turn that disbelief into curiosity. To be fair, Rudin offers an answer… “Part of the problem was that the concepts were not well-defined.” …though this is just another way of saying that they d’Almbert didn’t know modern math. But what was he thinking? Why did the objection seem natural to him? Why did it make sense? (There’s something sort of similar between some of the questions the math teacher and the math historian have to deal with.) The set of all singletons doesn’t exist From Classic Set Theory: Use appropriate axioms out of Zermelo-Franekel axioms 1-6 to show that $\{x: x \textup{ is a singleton} \}$ is not a set. I’m fascinated by sets that go wrong, and I was surprised that set of all singletons leads to trouble. Here’s my best attempt at explaining what goes wrong. The issue with “is a singleton” as the defining property of a set is that any set can be turned into a singleton via the Axiom of Pairing in ZF set theory. The Axiom of Pairing is like a machine: give it two sets, it’ll smoosh them together into one set. Give it the same set twice, and it produces a singleton. That way, there’s a singleton for every set. Slightly more rigorously, let $X$ be any set at all: “Hi, my name is $X$ and I’m a set.” Feed two copies of $X$ into the Axiom of Pairing; the axiom then spits out the set whose elements are just the elements of $X$ and, well, $X$: “Yo. The Axiom of Pairings made me. I’m the set whose elements are the same as those of $X$ and … err, ok, so just $X$. I’m $\{ X \}$. So, for any set X there exists a singleton set whose only element is X. Great! So what’s wrong with the set of all those singleton sets? The problem is another axiom, the Union Axiom. The Union Axiom is another little set theoretic machine. Feed the Union Axiom a set (chomp chomp) and it spits out another set. This new set contains all the elements that are in subsets of the original. An example is probably useful. Say that your set is the set of all the basketball teams in the NBA: {Bulls, Pacers, Knicks, etc.}. The Union Axiom is the set of all the players in the NBA. The Union Axiom bites through the husk of one collection and produces a new one out of the things living just one level down. Here’s where we start breaking things: what if you feed the set of all singletons to the Union Axiom? (Get ready, because here comes everything.) 1. The Axiom of Pairings can turn every set into a singleton. 2. The set of all singletons collects all of these singletons into a set. 3. The Union Axiom would create a collection of all the elements of elements of the set of all singletons, i.e. literally every set. 4. So the set of all sets exists. And that’s great, if you love contradictions, because now you can make any set you want, including Russell’s famous one. Because in this version of set theory the way that you block Russell’s Paradox is with the Axiom of Separation, which says that you can define a set using any property, as long as it’s a subset of some currently existing set. Now, though, we have a set of all sets. Everything can exist, bats explode out of the belfry, Pandora breaks the seal, boom! “Let there be a set that’s a subset of that set of all sets, containing all of the sets that are not members of themselves.” And that’s Russell’s paradox, and it leads to contradiction. All of which is a long way of saying, the set of all singletons does not exist. Which is weird, because I had written previous that this was the way Frege defined “1” within his system, as the set of all singletons. Is this set itself one way that the problems with Frege’s system manifested itself? Or was I misunderstanding Frege’s definition? One subtlety I’m missing out on right now is how the Zermelo-Fraenkel set theory system compares to other set theories. This bit from Wikipedia seems to summarize what I’ve just started to understand: Gottlob Frege and Bertrand Russell each proposed defining a natural number n as the collection of all sets with n elements. More formally, a natural number is an equivalence class of finite sets under the equivalence relation of equinumerosity. This definition may appear circular, but it is not, because equinumerosity can be defined in alternate ways, such as by Hume’s principle. This definition works in naive set theory, type theory, and in set theories that grew out of type theory, such as New Foundations and related systems. But it does not work in the axiomatic set theoryZFC and related systems, because in such systems the equivalence classes under equinumerosity are proper classes rather than sets. I’d like to understand some of the other sets that lead to trouble in ZF set theory. What sort of properties, in general, lead to contradiction? And I’d also like to understand what sort of choices you make in type theory in order to permit this sort of definition of “1.” “1” and Russell’s Paradox I’m falling in love with Classic Set Theory, an introductory set theory text. The introduction to Russell’s paradox motivates it in a way that I’d never seen before. When I’ve thought about Russell’s paradox in the past I’ve sometimes thought, ok, this makes sense, but how did he come up with this? Or, even, what line of thought could even have led to conceiving of something so clever? (Russell’s paradox is one of the great stories of math: a letter to Frege, just as he’s completing his master work, announcing a fatal contradiction in his system, effectively ending Frege’s life’s project. Wow! The above panel is from Logicomix, a book that I left feeling disappointed by. I felt it didn’t live up how awesome the story is, but maybe my expectations were just too high.) Anyway, here’s what I think I currently understand: Frege figured out a way to define “1” without talking about oneness. Instead, he made it all depend on uniqueness. Definition: A set $C$ is a singleton if there is an $x \in C$ and if anything else is in $C$, that thing is equal to $x$. Frege doesn’t say anything about the number “1” in defining singletons; it all depends on equality. So Frege is free to use the notion of singleton-ness to define the number “1”. Definition: There is a set of all the sets that are singletons. Call that set $[1]$. It’s a pretty nifty trick, and it shows that lurking within our notion of equality is some notion of oneness. So far, that’s Frege: “1” is a set, $[1]$. But if $[1]$ is a set, we can start playing around with that set. For example, can make a new set that contains just one element: $\{ [1] \}$ is a singleton! Which leads to the weird notion that ${ [1] }$ is a member of $[1]$. And, of course, $[1]$ is in${ [1] }\$ itself, so we get a weird series of inclusions:

$[1] \in \{ [1] \} \in [1]$

Not paradoxical yet, but weird. And close enough to Russell’s paradox that you can see where he might have started playing around with these ideas. OK, so there’s a weird way that $[1]$ as defined by Frege is a member of itself. How could we mess around with that idea some more?

Definition: V is the set of all sets.

And then clearly, V is also a member of itself.

And now we’re so, so, so close to Russell’s paradox itself: the set of all sets that doesn’t include itself.

Part of what I find so exciting about all of this is the connection to Frege, but also the sense that there’s a natural progression from the definition of “1” to the paradox. What it is is that the definition of “1” does open the door to weird inclusions, since it’s a set of everything that is a singleton. And once that door is open, everything can march on through.

The only way to close that door is to restrict what is allowed to become a set.