My superpower is synchronicity

I have often derided those who are susceptible to math envy. Y'know, the idea that math=intelligence. This utter foolishness leads to the simultaneous fear and awe of anyone who throws math around, as if the presence of mere symbols and equations demonstrates the clear superiority of the author's throbbing, bulging,... intellect. This utter foolishness leads, therefore, to authors who feel the need to add superfluous "mathematics" to their writings in order to demonstrate that their... intelligence measures up that of their colleagues.

Well, turns out, someone finally got around to doing a study on math envy: Kimmo Ericksson (2012) "The nonsense math effect", Judgment and Decision Making 7(6). As expected, those with less training in mathematics tend to rate utterly irrelevant "mathematical content" more highly than its absence. Lest anyone start feeling smugly superior, however, I'll note that I've seen this effect most strongly in those who should know better, i.e., those with just a little mathematical training. This includes, for example, computer scientists who are not formal theoreticians. Not to name names, but I've read more than one NLP paper that throws in some worthless equation just to try to look more worthwhile. (These papers are often fine, in and of themselves, but would have been better had they not succumbed to math envy.)

As Language Log points out in their coverage, this isn't limited just to math. Some people also have brain-scan envy and similar afflictions. That's definitely worth watching out for, but IME people seem more aware of their pernicious effects while being blind to math envy.

Last time we talked about sets which support a single binary operator. That's all well and good, but things get really interesting once we have two of those operators. Thus, for lack of a better name, I present a map of ring theory. The majority of this isn't actually part of ring theory proper, of course; but that's as good a name as any, since we'll want to distinguish these from lattice theory which also has two interlocking operators.

In the center of the map are (unital) rings. For those who aren't aware, in older literature, what are now called pseudorings used to be called "rings", and what are now called rings used to be called "unital rings" or "rings with unit". I've run into very few proper pseudorings of interest, so I support this change in terminology. In any case, if we start from rings and move upward we get rid of properties. In addition to pseudorings we also get semirings, which are utterly ubiquitous. In the next post I'll give a few dozen of the most common semirings, so I'll save my rant for then. If we keep moving up then we start getting rid of associativities and distributivities, resulting in things like seminearrings (aka near semirings) which arise naturally in certain kinds of parsing problems. This area is poorly explored, hence the dearth of names in the upper part of the map. This is, however, where I've been spending a lot of time lately; so you'll probably hear more about seminearrings and their ilk in the near future. As the names suggest, we have both left-near and right-near versions of these various structures, though I only draw the right-near ones for clarity.

Moving downward from semirings there are two directions to head. Off to the left we run into Kleene algebras and lattice theory yet again. And to the south we run into the swamp along the way to fields. In spite of their apparent axiomatization, fields are not really algebraic objects, which is a big part of the reason for all this mess. In the lower left we see a chain of inclusions based on some peculiar properties like every ideal being principal, the existence of greatest common divisors (though not necessarily computable with Euclid's algorithm), the ascending chain condition on principal ideals, etc. These properties will be familiar to the actual ring theorists, as would numerous other properties I didn't bother putting on here. Off to the lower right we get a different sort of breakdown. In particular, before we get unique inverses for all non-zero elements, we can instead just have pseudoinverses or strong pseudoinverses. This is similar to the distinction between (von Neumann) regular semigroups vs inverse semigroups.

There's plenty more to say about ring theory and related areas, but that would be a whole series of posts on its own. This is actually the first map I started to make, because this is the region where we find so many mathematicians coming in from different areas and not being familiar with all that has been done before. As I'm sure you notice, quite a lot has been done, both in breadth and in depth. I brought this map up once when chatting with Amr Sabry, and he's the one who convinced me to finally get around to posting all these. So, hopefully these'll give y'all a better lay of the land.

There are some notable omissions from this map as it stands. In particular, complete semirings (aka *-semirings) are missing, as are rings with involution (aka *-rings), as are the various constructive notions of fields (like discrete fields, Heyting fields, residue fields, meadows, etc.). Next time I'll talk a bit about complete semirings; they were omitted mainly for lack of space, but should be included in future versions. The various constructive notions of fields were also omitted mainly for space reasons. I'll probably end up making a close-up map of the swamplands between rings and fields in order to do justice to them. Rings with involution were omitted mainly because I'm not entirely sure where the best place to put them is. As the name suggests, they've been primarily studied in contexts where you have a full-blown ring. However, there's nothing about the involution which really requires the existence of negation. I've started chatting with Jacques Carette about related topics recently, though; so perhaps I'll have more to say about them later.

This map is released under Creative Commons Attribution-ShareAlike 3.0. Any questions? Anything I missed? Anything that needs further explanation?

Friday I turned in the last of my papers, and last night I turned in the last of my grades. Which means I'm free for a whole two weeks or so. Which is good because for the last couple weeks I've been wanting desperately to blog as a means of escaping all those due dates. So now I get to flush some of the backlog from this semester. In particular, there's a cluster of posts I've been meaning to put up for a while, a sort of Intro to Maths for mathematicians, as it were. To save y'all's friendspages I'll spread these posts out over the next few days.

columbicubiculomania — The compulsion to stick things into pigeonholes. (Jim Matisoff 1990)

Ever since stumbling upon that fabulous word I've wanted to spread its popularity. As a geek with certain obsessive–compulsive tendencies, I'm a bit prone to columbicubiculomania; or rather, as the theoretician that I am, I'm prone to the dual of columbicubiculomania. I don't care so much about the pigeons, they can take care of themselves. But I do have a certain compulsion to design and arrange pigeonholes such that whenever someone feels the columbicubiculocompulsion, they'll have the right places to stuff all their pigeons into. Part of this is also tied up in me trying to figure out (or rather, to convey) where exactly I situate myself in the vast sea of competing academic fields. As the perennial outsider, I'm more interested (it seems) in seeing how everything is connected than are those stuck on the inside.

And so, over the past few years I've been accumulating the jargon from different subfields of mathematics and organizing them into a cohesive whole. In particular, for the next few posts, I'm concerned with the terminology for algebraic objects. Too often I read papers with practitioners of one algebraic field (e.g., group theory, ring theory,...) getting unwittingly caught up in another field and subsequently using such outrageous terms as "semigroup with identity" or "monoid without identity". Because of this ridiculousness, a lot of mathematicians and computer scientists end up not realizing when the thing they're studying has already been studied in depth under another name. So let's see all those names and how they connect. Let's produce a cartography of mathematical objects.

Perhaps the easiest place to start is one of the latter maps I produced. Binary relations are ubiquitous in all areas of mathematics, logic, and computer science. Typically we don't care about all binary relations, we only care about relations with particular properties. Of course, the interaction between properties is nontrivial since properties A and B together can entail that property C must hold. Thus, a map of binary relations is helpful for keeping track of it all. This map requires relatively little explanation, which is why I started here.

All the common properties of interest are defined at the top, and color coded for use in the map. And, constructivist that I am, I've been sure to distinguish between strong and weak versions of the properties (which collapse in the classical setting). The arrowheads in the map are for keeping track of when we're talking about P, anti-P, or co-P (for some particular property P). And the big black dot is the starting point of the map (i.e., a binary relation with no special properties). The dashed lines indicate when some property will follow for free. So, for example, there's no name for a transitive irreflexive relation since trans+irrefl entails antisymmetry and trans+irrefl+antisym is a strict partial order. And the black lines are for when the named object requires some peculiar property that doesn't show up all over the place. These conventions for dashed and black lines are common to all my maps.

Once you're familiar with my conventions, the only thing really left unstated on this map is that the apartness relation used in the definition of a strongly connected binary relation is actually a tight apartness. The constructive significance of apartness relations vs equality relations is something I'll have to talk about another time.

Towards the top of the map we start veering away from binary relations per se and start heading into order theory. Because the complexities of order theory are a bit different from the complexities of binary relations, I've chosen to break them up into different maps. As yet, I haven't actually made the map for order theory, but I'll get around to it eventually. Lattice theory and domain theory also gets tied up in all this (in the order theory particularly). I've started work on a lattice theory map, but haven't finished it just yet.

This map is released under Creative Commons Attribution-ShareAlike 3.0. Any questions? Anything I missed? Anything that needs further explanation?

The integers mod 3 are better thought of as {-1, 0, 1} rather than the traditional {0,1,2}. This makes it clear that the two non-zero elements are additive inverses of one another. And it also makes it clear that multiplication of non-zero elements just returns an "is same/different" judgment, which comes for free from our usual arithmetic: (-1)*(-1) = 1 = 1*1 and (-1)*1 = -1 = 1*(-1).