Singmaster has been described as “one of the most enthusiastic and prolific promoters of the Cube”.

Some writers, such as David Joyner, consider that for an algorithm to be properly referred to as “God’s algorithm”, it should also be practical, meaning that the algorithm does not require extraordinary amounts of memory or time.

disconcision replied to your post “ There’s something a little mysterious to me about the usage of “the”…”

aliens guy: “category theory”

i feel like what you’re getting at is that we can fall into an intuitive sense that equality of objects lives in some kind of objective arena. what we find out in math (or just thinking precisely about anything, i guess) is that this is abjectly not the case. equality is always with respect to some underlying category/type/whatever, and which arena we choose as our ground determines which things are equal and hence what is unique

also reminscent (though tangentially): http://math.ucr.edu/home/baez/qg-spring2004/s04week01.pdf

Ooh, yeah, those Baez notes seem to be talking about the same thing as my “constraints” and “structure” (except he says “properties” instead of “constraints”).

Anyway, your second reply feels right, but I think there is a little more here than just “equality of objects is relative” – that sounds like an observation about some independently defined things called “objects,” as if we have a good handle on what an “object” is but not necessarily on when objects are equal.  But it’s actually that way of looking at things that (in various specific cases) tends to feel wrong to me.

It feels like there is a tension between two ways of thinking which are both supposed to be hallmarks of modern/higher math: formalism and abstraction. Formalism tells you that the explicit capital-D Definition of an object is the ultimate source of truth about it, and closer to “what the object really is” than the set of the motivating examples you keep around in your head and use for intuition. But abstraction tells you to care only about intrinsic patterns/structures and not the contingent ways they may happen to be encoded. From the perspective of abstraction, a formal definition is just a way of expressing a pattern, and our intuitions can get at aspects of the pattern that the definition misses. (E.g. if we move away from set-theoretic foundations, no one is going to say that the word “group” can’t be used anymore because a group just is “a set equipped with (etc).”)

To continue on this riff: formalism has this problem where it allows you to start with definitions that have more structure than you really want, and then happily carry it around with you forever, expressing its irrelevance by saying “two objects that differ only in that way are isomorphic” — as if this is some further fact you’ve happened to learn about the pattern you are studying, when it’s really a fact about your (bad) encoding. For example, for any kind of object based on a set, we could imagine forming a stupid variant of that object where the set is ordered (a tuple), and then all of the results would be the same except we’d pointlessly act like there were multiple copies of each instance (one per ordering) that “just happen to be” isomorphic. I don’t think anyone does exactly this, but there’s this uncomfortable feeling that the same kind of error could be happening in fancier ways without us noticing.

Two things I think are kind of interesting about this:

(1) I feel like programming computers has shaped the way I think about this stuff. I’m used to drawing the distinction between the data I want to store and the data structure / encoding I use to store it, since the former is usually fixed by the problem at hand but the latter can be chosen and matters for speed, etc. So, in programming, formalism — taking one way of encoding the data and saying it is the data — is recognizably a bad habit, which will prevent you from finding better encodings. (The set vs. tuple thing from the last paragraph is a common practical issue in my everyday work, that’s why it came to mind!)

(One could imagine a parody of formal mathematics in which every definition starts by telling you that the object is stored as JSON.)

(2) It’s kinda cool how this tends to connect very abstract and very down-to-earth ways of looking at something, distinguishing them from some middle ground. You start out with an intuition that something can be abstracted from some examples, then you write a formal definition of the abstraction, but then as you prove more equalities / isomorphisms, you find more and more ways that your “naive” original examples are completely representative of other things, while the formalism can get more and more (but not less) misleading.

Arnold mentioned a few cases like this in “On Teaching Mathematics”:

What is a group? Algebraists teach that this is supposedly a set with two operations [two?? -nost] that satisfy a load of easily-forgettable axioms. This definition provokes a natural protest: why would any sensible person need such pairs of operations? “Oh, curse this maths” - concludes the student (who, possibly, becomes the Minister for Science in the future).

We get a totally different situation if we start off not with the group but with the concept of a transformation (a one-to-one mapping of a set onto itself) as it was historically. A collection of transformations of a set is called a group if along with any two transformations it contains the result of their consecutive application and an inverse transformation along with every transformation.

This is all the definition there is. The so-called “axioms” are in fact just (obvious) properties of groups of transformations. What axiomatisators call “abstract groups” are just groups of transformations of various sets considered up to isomorphisms (which are one-to-one mappings preserving the operations). As Cayley proved, there are no “more abstract” groups in the world. So why do the algebraists keep on tormenting students with the abstract definition? […]

What is a smooth manifold? In a recent American book I read that Poincaré was not acquainted with this (introduced by himself) notion and that the “modern” definition was only given by Veblen in the late 1920s: a manifold is a topological space which satisfies a long series of axioms.

For what sins must students try and find their way through all these twists and turns? Actually, in Poincaré’s Analysis Situs there is an absolutely clear definition of a smooth manifold which is much more useful than the “abstract” one.

A smooth k-dimensional submanifold of the Euclidean space R^N is its subset which in a neighbourhood of its every point is a graph of a smooth mapping of R^k into R^{(N - k)} (where R^k and R^{(N - k)} are coordinate subspaces). This is a straightforward generalization of most common smooth curves on the plane (say, of the circle x² + y² = 1) or curves and surfaces in the three-dimensional space.

Between smooth manifolds smooth mappings are naturally defined. Diffeomorphisms are mappings which are smooth, together with their inverses.

An “abstract” smooth manifold is a smooth submanifold of a Euclidean space considered up to a diffeomorphism. There are no “more abstract” finite-dimensional smooth manifolds in the world (Whitney’s theorem). Why do we keep on tormenting students with the abstract definition?

breaking my self-imposed tumblr break from non-queued posts to say thank you to this blue hellsite

birdblogwhichisforbirds:

I’m thankful for this blue hellsite. I’m thankful for a lot of things and listing them would probably take too long to list, but I wanted to share this.

I mean it.

There are a bunch of justifiable complaints about this site’s poor functionality and about the bad discourse and misinformation spread by its users. This site has been bad for my mental health in various ways and at various times. I’ve deleted multiple blogs and remade. But I’m thankful.

I am beyond grateful that in the far off days of 2011 I came on this website to post about Sherlock and Doctor Who. I am thankful that when I was throwing up everything I ate and chronically suicidal and convinced I was going to Hell for being bi, I was confronted with the annoying rainbow-spewing of 2011 tumblr. It was cringe and imperfect and fetishizing but it was profoundly healing. And I met my first girlfriend through this website and I gave up Catholicism for Lent.

Despite the absurd pettiness of some of it, I am thankful I engaged in fandom. I am thankful that in a period of my life when I felt utterly hopeless about most things, I was able to be passionate about a mad man in a box. I am thankful that, while my self-confidence and sense of my values was completely shattered and I didn’t know what my life mean without my faith, I was able to be strident about defending my favorite TV show. It wasn’t the healthiest coping mechanism, but it was something and I needed it.

I’m thankful I read disability tumblr. I’m thankful I learned things about myself and others that I otherwise wouldn’t have. I am thankful that it told me that some of the things I had to do in First Ever Job (at a residential school for disabled teenagers) were wrong. I am thankful that I met people who understood chronic suicidality and who helped me crawl out of it. I am thankful that I was able to find techniques to get my brain to Actually Do Stuff. I am thankful that, when I didn’t have access to good mental health care (because NHS mental health is underfunded and I had no money for private therapy) I could get internet advice that kept me alive.

I am beyond grateful for rationalist/EA tumblr. I am beyond grateful for being able to find a way to values that were meaningful to me after God. I am so lucky to have learned about better ways of doing good, and better ways of thinking. I have found a community that loves and accepts me and makes me a better person. I have had issues with this community, I have had one very unpleasant break up inside it and I think some things have fueled my scrupulosity. But this community has given me tools I needed to build  life that has value to me, and I don’t know how I would build that life without it. I have met so many wonderful people, including my husband.

I am so thankful for Rob. I am so thankful for his love an his patience and his kindness. I am so thankful that we “get” each there (most of the time) without it being a constant uphill struggle. I’m gonna quote from the beginning  of the speech at our wedding (I have it saved on my laptop):

Now this is a story all about how my life got flipped, turned upside down, and if you’d like to take a minute just sit right there I’ll tell you how I moved four and a half thousand miles for a boy from the internet.

On 20th June 2014, I was wasting time on tumblr dot com and I saw a long post about ethics. It was by someone with the username nostalgebraist who I’d seen arguing with various friends of mine about Bayes’ theorem and other statistical questions, but I hadn’t followed him because I didn’t know enough Mathematics to make sense of it. But in this post, he said “One can’t build a moral system from hating oneself, because one changes; once one decides “this is Good because it is not what a wretch like me would do,” it thus becomes something a wretch like you would do, and thus it flips back to being Bad again. […] If you mock me for playing around in my academic sandbox when I could be thinking about bigger grander things, you are saying that I should be thinking about how I am fundamentally disgusting scum and how God hates me (I don’t believe in him, but he hates me anyway).” And I thought, oh! A wretch like me! WE CAN BE WRETCH-BUDDIES! So I followed him, and I talked to him sometimes, and we discussed our mutual wretchedness. We chatted with each other on tumblr a lot over the next year or so, and as time went by we got closer and started talking to each other outside of tumblr as well.

And from the end of that speech:

For months and months while we were waiting for me to get my visa, I used to say to Rob “I’m gonna marry you so hard.”  And now I’ve done it and I’m double-marrying him, because once was not enough. I was scared, before I came here, because I knew I was taking a risk. But the alternative to moving halfway across the world was an even bigger risk. I would be risking waiting the rest of my life to find someone as wonderful, and as compatible with me, as my Rob. I don’t think that would have worked out. And now, I am here and I am glad I took this risk. I am glad to be in this beautiful city and to have been welcomed by the lovely community of people I’ve met here. I am glad that people from outside have travelled really long distances to come and support us, and many many others who couldn’t be here have sent their love and good wishes. But most of all I am glad that I found someone to be wretch-buddies with and that over the course of several years, we have stopped being wretches. I am not the wretch I was in 2014 and I think a large part of the credit for that goes to Rob. And he is not the wretch he was in 2014 and I hope that some of the credit goes to me.

Rob is an incredibly careful person. Rob is careful because he is full of care both in the sense of worry and in the sense of compassion. He has treated me with so much care over the last four years, and I cannot thank him enough. But despite his carefulness, despite his natural aversion to risk, he took a chance on me. I am so glad that he did. I am so glad I am here. I am so glad to be getting married.

Also, I know this sounds shallow but I don’t fucking care: I am grateful that I am not (first-world definition of) poor anymore. I am grateful I don’t have to go into debt to buy food. I am grateful that I don’t have to eat expired food I stole from the bins at work (and not tell anyone, because there were people who could have helped me but I was ashamed.) And I think I would have made my way out of that without Rob - I was doing better financially than ever before in the last year or so in England but like… Rob helped. He thinks it is normal to not be in debt. He thinks it is normal to have savings. And not just like, two hundred pounds in savings. He thinks it is normal to have a retirement plan. And if he’d been poor but we had somehow found a way to afford immigration despite that, I still would have married him. But like… if you want to improve your mental health, marrying a tech bro and never having to eat literal garbage again really doesn’t hurt.

I feel like I spent the first half of my 20s digging a hole for myself and the second half crawling laboriously out of that hole, while backsliding frequently. And I think I blamed my overuse of the internet and of this website for that, and there was some truth to that. But I think it was also my way of getting out of that hole.

But I think I had a lot of needs that I couldn’t really get met any other way. And I don’t think there were any other ways that were accessible to me. I think that getting those needs met in ways that were better would have acquired attention span/ money/ energy/ executive function/ proximity to specific meatspace communities/ self-trust I didn’t have then. And now I do have those things, and I don’t need this blue website so much any more (which is a good thing, because one of these days Yahoo! is gonna stop their half-hearted attempts to make money from it and pull the plug, and that could happen in 30 years or it could happen next week.) But I am thankful to people I met here.

Back in 2011 people would say “tumblr teaches me more than school!” (because tumblr would tell them things like “being trans is a thing” or “the clitoris is a thing” or “racism is bad” or “here is a somewhat over-simplified map of native American tribes” or “you should feed your pets good quality pet food and not random garbage” and apparently their schools were failing to impart this extremely basic stuff.) Other people would make fun of this as cringey and bad, especially as this site is full of bad opinions and  misinformation. But this website has taught me things I am genuinely grateful for. And that is thanks to the people I met here. Some of them have left, one of them I fell out with even though I am still incredibly grateful for some of the things they taught me. But all of them helped me.

Thank you, terrible discoursey broken wearing-a-tie-as-a-belt website. Thank you people on here.

<3 may you all have as much to be thankful for as I do.

Eerily, poignantly appropriate that (1) I experienced a tumblr glitch immediately after liking this post and (2) that it was this specific glitch

the-moti:

nostalgebraist:

 There’s something a little mysterious to me about the usage of “the” vs. “a/an” in math.  It seems related to a difference which comes up when we’re characterizing mathematical entities through their properties:

1. Sometimes we want to make statements that apply to every thing that has these properties, even things that also have some other properties we haven’t mentioned (”a/an”)
   
2. Sometimes we’re trying to single out an object characterized by these properties and nothing else (”the”)

After thinking about it for a while recently, I get the sense that you can look at a lot of things in either of these ways, and the standard linguistic choice just reflects the perspective that comes more naturally, not some specific type of property that’s shared across every case.  But maybe I just don’t understand this?

The difference between (1) and (2) is that (1) applies after adding properties to an object.  By “properties” I’m actually thinking of two different kinds of things – I’ll call them constraints and structure.

Constraints are extra equations of the same kind as the original characterizing ones.  When you characterize a group by its presentation, you specify the (cardinality of the) underlying set along with some equations relating an element to another  So, for example (thanks Wikipedia), the cyclic group C_8 has presentation < a | a^8 = 1 >.  But this doesn’t just mean that it has one element, a, satisfying the equation a^8=1 – because there are another groups like C_4 and the trivial group that satisfy this equation.  What uniquely identifies C_8 is that it is the “freest” object fitting this description, i.e. the one that doesn’t satisfy any other equations.

Some of the things that can be said about C_8 would be equally true for any group with one generator satisfying a^8 = 1, and we could imagine having a (similar but not identical) description of these things.  We would call these “C_8s” or some other plural noun, we would say things like “a C_8,” and the specific group now called C_8 would be “the free C_8.”  This is the situation for the relations that characterize Abelian groups, for instance.  (The reverse would be to call the free Abelian group “the Abelian group” and call specific Abelian groups “homomorphisms of the Abelian group” or something.)

Structure, as opposed to constraints, means properties of a different kind which are invisible from the perspective of the original characterizing properties.  With a group, you can turn it into a ring by adding another operation, but this is not related to group-level properties (i.e. not relevant for group isomorphism): you can’t look at a group and say whether it’s “currently being a ring operation” rather than “just being itself,” the way you can say whether or not something is the freest group of some description.

The justification for collapsing all objects fitting a description into a single object, worthy of “the,” usually involves some particular isomorphism.  All of the objects satisfying the (absolute) presentation of C_8 are group isomorphic, so from the group perspective, it feels like there’s just this one thing, C_8.  But you can of course exhibit two different versions of C_8 with some extra structure, and don’t have that structure’s isomorphism.  In this way, you can make any one thing plural by adding some extra distinguishing variables.  So “the” is always at risk of turning into “a/an” if you find some companion structure you want to talk about a lot.

Like, why do we say “a vector space of dimension n over R,” rather than “the vector space of dimension n over R,” since they’re all the same thing (isomorphic)?  This was the thought that led me into this – that’s always felt off to me somehow.  And it seems like the reason is that these objects (pretty boring by themselves) are mostly used with extra structure added, so it’s natural to think of there being many different versions of each one.  (This is equally true whether it’s something like an inner product, or something about what the vectors “really are,” e.g. the polynomial vector space P_2 and Euclidean 3-space have the same dimension, but you can evaluate polynomials at points in R, you can’t do that with Euclidean points.)  This is very different from the situation in group theory, where you are thinking of groups abstractly and isomorphism feels like identity.

This perspective also seems to illuminate why I always found descriptions of vector space duality weirdly offputting.  They’re talking about these two “different” vector spaces, but they’re isomorphic, so in pure vector-space-world, what difference could they have?  I guess the answer is, each one of them is actually given some (different) extra structure.  But this extra structure is described entirely in terms of vector spaces and it’s easy to get the sense it is something intrinsic.  (I guess I am saying that once you are talking about V and V*, these objects are no longer quite as generic as pure vector spaces.)

I think your intuition can be made completely precise here. Saying “a vector space of dimension n” and therefore pretending that there are many different vector spaces of dimension n is a hack to help our brains avoid thinking about non-canonical isomorphisms. If you say “Let V be a vector space and let V^v be the dual vector space” then anything you say after will be invariant under the action of GL_n on V. If you consider a vector space R^n, you may view it as both a space and the dual space, and thereby construct something that is only invariant under O(n).

So some mathematicians will believe that we should only say “the” when the object in question is unique up to unique isomorphism.

(via the-moti)

 There’s something a little mysterious to me about the usage of “the” vs. “a/an” in math.  It seems related to a difference which comes up when we’re characterizing mathematical entities through their properties:

1. Sometimes we want to make statements that apply to every thing that has these properties, even things that also have some other properties we haven’t mentioned (”a/an”)
   
2. Sometimes we’re trying to single out an object characterized by these properties and nothing else (”the”)

After thinking about it for a while recently, I get the sense that you can look at a lot of things in either of these ways, and the standard linguistic choice just reflects the perspective that comes more naturally, not some specific type of property that’s shared across every case.  But maybe I just don’t understand this?

The difference between (1) and (2) is that (1) applies after adding properties to an object.  By “properties” I’m actually thinking of two different kinds of things – I’ll call them constraints and structure.

Constraints are extra equations of the same kind as the original characterizing ones.  When you characterize a group by its presentation, you specify the (cardinality of the) underlying set along with some equations relating an element to another  So, for example (thanks Wikipedia), the cyclic group C_8 has presentation < a | a^8 = 1 >.  But this doesn’t just mean that it has one element, a, satisfying the equation a^8=1 – because there are another groups like C_4 and the trivial group that satisfy this equation.  What uniquely identifies C_8 is that it is the “freest” object fitting this description, i.e. the one that doesn’t satisfy any other equations.

Some of the things that can be said about C_8 would be equally true for any group with one generator satisfying a^8 = 1, and we could imagine having a (similar but not identical) description of these things.  We would call these “C_8s” or some other plural noun, we would say things like “a C_8,” and the specific group now called C_8 would be “the free C_8.”  This is the situation for the relations that characterize Abelian groups, for instance.  (The reverse would be to call the free Abelian group “the Abelian group” and call specific Abelian groups “homomorphisms of the Abelian group” or something.)

Structure, as opposed to constraints, means properties of a different kind which are invisible from the perspective of the original characterizing properties.  With a group, you can turn it into a ring by adding another operation, but this is not related to group-level properties (i.e. not relevant for group isomorphism): you can’t look at a group and say whether it’s “currently being a ring operation” rather than “just being itself,” the way you can say whether or not something is the freest group of some description.

The justification for collapsing all objects fitting a description into a single object, worthy of “the,” usually involves some particular isomorphism.  All of the objects satisfying the (absolute) presentation of C_8 are group isomorphic, so from the group perspective, it feels like there’s just this one thing, C_8.  But you can of course exhibit two different versions of C_8 with some extra structure, and don’t have that structure’s isomorphism.  In this way, you can make any one thing plural by adding some extra distinguishing variables.  So “the” is always at risk of turning into “a/an” if you find some companion structure you want to talk about a lot.

Like, why do we say “a vector space of dimension n over R,” rather than “the vector space of dimension n over R,” since they’re all the same thing (isomorphic)?  This was the thought that led me into this – that’s always felt off to me somehow.  And it seems like the reason is that these objects (pretty boring by themselves) are mostly used with extra structure added, so it’s natural to think of there being many different versions of each one.  (This is equally true whether it’s something like an inner product, or something about what the vectors “really are,” e.g. the polynomial vector space P_2 and Euclidean 3-space have the same dimension, but you can evaluate polynomials at points in R, you can’t do that with Euclidean points.)  This is very different from the situation in group theory, where you are thinking of groups abstractly and isomorphism feels like identity.

This perspective also seems to illuminate why I always found descriptions of vector space duality weirdly offputting.  They’re talking about these two “different” vector spaces, but they’re isomorphic, so in pure vector-space-world, what difference could they have?  I guess the answer is, each one of them is actually given some (different) extra structure.  But this extra structure is described entirely in terms of vector spaces and it’s easy to get the sense it is something intrinsic.  (I guess I am saying that once you are talking about V and V*, these objects are no longer quite as generic as pure vector spaces.)

science-of-noise:

Time Traveler from 1919: You have a day where you honor our struggles in the Great War, the War to End All Wars?

Me: You mean World War I?

TT: Oh shit

(via dharma-initiative-official)

homo-j-simpson:

you think youre a nerd cuz you play dungeons and dragons?? well, the REAL nerds cant even find people to play dnd with

(via sua-sponte-deactivated20200331)