There’s an interesting article in Wired. It’s difficult to summarize. To me it seems silly on the surface, but I’m a mathematical lightweight.
2+1 and 3 seem to be equal in quantity. Seems like sets of objects can have more properties than just quantity though.
That article was fascinating. It all sounds weird and arbitrary and confounding, but the process matches exactly much of what I’ve read about the history of math.
That’s littered with major advances by geniuses which required acolytes who could act as go-betweens and make the abstruse accessible. It’s always the next generation who grows up learning the new stuff without having to forget the old that builds of the new to advance the field into new territory.
Schoolkids aren’t going to need to worry about losing the equals sign. Equivalence is way beyond simple algebra. Most of modern math isn’t even translatable into English terms that non-specialists can follow. That’s bad for mathematicians in one way, because a field is always deprecated when no one has any notion of what they do. But it’s good for mathematicians in another way. They used to have to worry about cranks proving it was possible to trisect an angle or that 0.9999~ isn’t equal to 1. But nobody is going to try to inflict their theories about Higher Topos Theory. They won’t get past the title.
To use an example, 2+2=4 is a common equality. But suppose I’m looking for pairs of prime numbers? In that case, 2+2 is a valid answer but 4 is not; so they are not equal for my purposes. But I would accept that 2+2=3+3.
Or to use a less abstract example, five pennies are equal to one nickel - unless you’re replacing some blown fuses.
That sounds like a five-year-old’s logic: “Four is bigger than three, right? But what if the four is small and made of wood and the three is big and made of stone?”
??
It’s really not that complicated (except when it starts getting highly technical). Consider, for example, a finite-dimensional vector space, the same thing introduced in a first-semester algebra course. Let’s call it V. Next, consider the space of linear homomorphisms V → F, where F is the field of scalars. This is the “dual space” V*. Next, you can form V**. If you try to unwind the definitions, this double dual looks complicated and confusing, consisting of maps into a set of maps. However, V** is naturally/canonically isomorphic to V, and you may as well write V = V**, even though at first glance it might seem they are not the same.
In other words, you can realize the same structure in several ways. Similarly when you study abstract algebra: now you have categories of all sorts of algebraic structures. But two different categories may be naturally equivalent to each other. (This might be really interesting, since the categories might arise in totally different ways.) More complicated algebraic structures might require the use of “higher category theory” as described in the article.
That’s good to know
Yeah.
Sorry, DPRK but I’m afraid you didn’t do a very good job of explaining this “really not that complicated” matter :D.
finite-dimensional vector space: The one you are most familiar with is just the set of points in 3D space. How do you specify a specific point in your room? It has a height, a distance from one wall, and a distance from a right-angle wall. That’s three coordinates (the finite part). The “space” is the set of all points that you can describe this way.
linear homomorphisms V → F: Suppose you’re using your new coordinate system to describe the temperature in your room. For each point, you can use your thermometer to record a single value (a scalar). Suppose also we know that the temperature varies in a specific way: in particular, we know that the temperature can always be described as Aheight+ Bwall1_dist + C*wall2_dist. Mathematicians call this a “linear” map.
So we ask: what are all the different ways that we can map a point to a temperature (limiting ourselves to linear maps)? Or put another way, what are all the temperature distributions that our room can take on that can be called linear? Well, it turns out that the “space” of all these ways has the same number of dimensions as in the original vector space. That’s not too hard to see here: the constants A, B, and C controlled the behavior of our map. There was one constant for each coordinate in the original vector.
Slightly trickier: what happens when we do this operation twice? What is the space of all the different spaces that describe how to map a coordinate to a temperature?
It’s really hard to wrap your head around what that means, but it doesn’t matter, because we know again that it will have three coordinates. Finding the “dual” didn’t change the number of coordinates the first time, and it won’t change the second or hundredth time either.
But there’s another aspect that’s trickier to describe. I’ll go with a simpler example: finding the dual space is a little like finding a number that, when multiplied by the original, equals 1. So, if we start with 3, then our dual number is 1/3, because 3*1/3=1. What’s the dual-dual of our number? It’s just the original! 3 goes to 1/3, which goes to 3.
When you go to higher dimensions, it’s a little more complicated, but the basic principle is the same. Starting with some object E, we find a kind of inverse object E’ such that E*E’ = 1. And the same thing works in reverse, because E’*E = 1 too.
That’s what DPRK means then when he says that V = V[sup]**[/sup], at least in some sense. The descriptions were totally different, one being a fairly common-sense thing and the other being a weird map of a map. And yet they’re provably equivalent.
(hope I didn’t screw this up too much, so I welcome corrections)
Define “up”.
Mea culpa
Actually, to give an example of two objects that look different but are, in an appropriate sense, “the same”, does not take linear algebra (but I thought an algebraic example was appropriate since this is all heading in the direction of abstract algebra).
For instance, in the world of sets you need to somehow define what are ordered pairs, like (a, b). One way to do that is to define (a, b) as { {a}, {a,b} }. But you could have just as well defined it as { {b}, {a,b} }. It looks (and is) different, but is the same thing as far as anyone is concerned.
In fact, in abstract algebra, one is concerned not merely with “objects” (e.g., sets), but rather with categories of objects (sets plus functions between sets). So you have the category of finite sets, the category of all sets, the category of finite-dimensional vector spaces, the category of groups, and so on.
Back to Dr. Strangelove’s point: if you have two objects in a category, say A and B, sometimes they are isomorphic in the sense that there is a map from A to B, call it f, and also a map g from B back to A, such that f and g are inverses of each other, so gf = fg = 1. In the case of sets, this simply means that A and B have the same number of elements. But if you have two different categories, say C and D, then we want to say when they are the same- this is called “equivalence”- so you will have F: C → D and G: D → C, except now it is not true in general that G(F(x)) = x and F(G(y)) = y, instead, for any x, G(F(x)) will, in general, only be “naturally isomorphic” to x in the sense I started describing for ordered pairs and for double duals of finite-dimensional vector spaces.
The problem now is, you may need to consider categories with additional structure, which amounts to categories of categories, and check whether they are equivalent, and so on to higher levels. Even to write down the set of diagrams and equations such an equivalence must satisfy quickly becomes unwieldy (we are talking something like 50 pages of diagrams just to define a 4-category) and then impossible, which is why the kind of topology-inspired technical machinery introduced by Lurie and others is indispensable.
Yes; the point was that (assuming V is finite-dimensional) V and V** are not only isomorphic (note that V and V* are already isomorphic, but to write down an isomorphism requires choosing some coordinates), but naturally isomorphic.
That sort of thing becomes important once you consider the “category” of all (say, finite-dimensional) vector spaces, or, more abstractly, any category, and you want to define what it means to be equivalent to some other category. Technically, this means that you have maps (functor) F: C → D and G: D → C, but now FG and GF are not the identity, they are only naturally equivalent to the respective identity functors.
SUMMARY VERSION: if you are doing abstract algebra, your objects are not concrete and you can’t really say when two things are “equal”, only that they are equivalent in an appropriate sense, therefore the equality sign gets a bit abused.
Just use some new symbols. We had to do that in OO programming decades ago. The equal sign is fine, don’t blame the symbol if you are using it the wrong way.
The only part of that which is explained is the “linear” bit, I’m no clearer on what “homomorphisms” are, nor what the V, the arrow and the F mean.
You may be using terms that aren’t in commonly understood by non mathematicians. If it is impossible to explain in simpler terms then that’s fine, but I’m not learning much from your explanations as currently stated.
In fact, looking back at the original extract quoted by davidm I think I got the concept pretty well from that and I’m not understanding what you are saying well enough to know whether you are even for or against it!
You have “objects”, like x, and you have “maps”, like x [sup]f[/sup]→ y. These satisfy various axioms; for example, if f: x → y and g: y → z, then you can compose them to get a new map gf: x → z, and this composition is associative, so h(gf) = (hg)f. (This is for now really an equals sign, since maps between two objects are supposed to be elements of a set, and in set theory there is the notion of whether two elements are the same or not.)
In linear algebra, vector spaces are the “objects” and linear maps are the “maps”, aka homomorphisms.
The thing is, with just this level of category theory, it is easy to keep track of all these axioms and the complexity requiring “higher category theory” does not yet emerge. You do not need it to do basic linear algebra. We do already see some abuse of notation, as it were, when people write things like V = V**, though. In fact, from the linear algebra point of view, considering the “dual” of a vector space and the “tensor product” of two vector spaces does bump you one notch up the “higher category theory” ladder when you try to enumerate all the different coherence conditions these now have to satisfy. Just a little, though.
In general, a higher-level category will not just have objects and maps between objects, but also maps between maps, and maps between maps between maps… it’s not something one would seem to encounter studying linear algebra, but this sort of thing, but where you have to go to infinity, does seem to arise naturally in the category of topological spaces, homotopy theory and situations like that.
Or maybe I am misinterpreting the thrust of the article, I don’t know…
Right. Despite the article’s provocative title, non-mathematicians aren’t going to change the way they do basic arithmetic and algebra or stop using the = sign. And even in basic “math-major math,” the notion of equivalence relations is well-established and non-controversial, so the idea that mathematical entities can be equal-ish without being exactly the same is nothing new. (It’s not new in computer science either—eg. Javascript’s distinction between == and ===.)
Apparently, what the article is talking about is something new; but whether it will catch on and turn out to be important, I cannot say.
But in some cases it’s true. Suppose you have a small quantity of paint on your brush and you’re painting a number. You might have just enough to paint a seven. But you might run out if you try to paint a six. Six takes more paint than seven even though it’s a smaller number.
If I’m understanding the concept correctly, what it’s saying is that the equal sign works when you are acting under a certain set of assumptions. Which is fine; that set of assumptions is perfectly valid. But the article is pointing out that there are cases where the assumptions are not true and they should also be considered.
It’s like how classical geometry was based on the assumption that everything was happening on a two-dimensional plane. It worked just fine. But in the nineteenth century, some mathematicians began exploring how geometry would work on a different type of surface.
Thudlow Boink’s example of == vs === is right on, illustrating the difference between equality and equivalence.
But (feel free to correct me if I’m wrong!) I don’t think this is a matter of classical versus non-classical, more like moving from the rigid world of sets, where two sets are either equal or not equal, to richer structures, where this issue comes up. This Lurie stuff is related to algebraic topology, if I understand correctly, where spaces or functions may not be exactly the same but may be deformable into each other. Higher category theory also comes up in physics as topological quantum field theory, also naturally in abstract algebra as soon as you consider tensor products (at least as a baby example). Anyway it seems to crop up in various contexts, so it’s good to know it when you see it.
There are other complex relationships in data processing. Object A and object B may be absolutely identical in every way, possibly they are the same object, yet they may be two different objects. It can get more complex than that as subobjects take you down the rabbit hole. There can be many different types of equivalences between objects and I’m sure mathematicians face the same situation. You just can’t cram all those different kinds of relationships in one or two symbols.