LaTeX

It seems the LaTeX engine I use on this blog is broken... I'll be replacing it soon, assuming it doesn't fix itself first.

Structuralist philosophy and methodology in mathematics

There is a philosophy of mathematics (or rather a collection of related philosophies) called structuralism. In brief, a structuralist believes that mathematical "objects" are positions in a structure, rather than existent objects. This is a rather incohesive and shallow ramble about structuralist methodology and philosophy in mathematics.

Since I hold a rather formalist, and somewhat classical platonic view of mathematics-- which is to say, I do not believe mathematical notions exist in the real world in any real capacity, but rather in some external abstract universe-- I intend to talk about brands of structuralism which do not in any way invoke "reality". Perhaps mostly because of my biases against "the real world", I cannot rightly fathom philosophies of mathematics which invoke the real world in any way.

Anyway, it seems that structuralism as a methodology pre-dates structuralism as a philosophy. What is a "structuralist methodology"? It is the approach which emphasises structures over systems. To use some language from logic, a structural methodology approaches theories, instead of models. A simple example is the tacit tendency to forget the difference between isomorphic groups: $\mathbb{Z}/3\mathbb{Z}$ is "the same group" as $\mathbb{Z}_3$. From a purely set-theoretic or material point of view, this is not correct: the first group has cosets of $3\mathbb{Z}$ in $\mathbb{Z}$ as group elements, and the second has $\{1,2,3\}$ as its underlying set. But the two groups are isomorphic, which means that they act the same as groups The tendency to forget the (quite irrelevant) difference between the two groups is the heart of structuralist methodology.

The Bourbaki group was one of the first to emphasize "high abstraction". Their methods are truly similar in spirit to the modern category-centric structural approach. While I have never read Bourbaki, all the information I can find leads me to believe that the set-theoretic foundation is a result of 2 things: when Bourbaki started, set theory was the only thing to work with (categories had not yet been invented), and "Bourbaki is relentlessly linear in its exposition". With this linearity in mind, changing to a categorical perspective late in the game was out of the question.

The structural approach permeates mathematics, particularly in algebraic areas, and in almost all contemporary approaches to foundations. Isomorphic structures are taken as identical; in category theory (and especially in higher category theory), there is a real push to eliminate notions which do not interact sensibly with equivalence-- equivalence is a weaker notion than isomorphism, but it is still considered a "good enough" notion of equality.

With the ubiquity of structural methodologies in mind, it should be no surprise that a closely related philosophy should spring up. I'm only surprised that it took so long (at least 40 years from the start of Bourbaki) to really pin down "structuralism". A structuralist philosophy takes this methodology as a philosophical starting point: it is not simply productive to study mathematical ideas from a structural viewpoint, but mathematical objects are structures. 3, for example, is not a specific set (e.g. {{}.{{}}.{{}.{{}}}} or {{{{}}}}), but rather a convenient short-hand for any object satisfying a "3-like" position in a structure. This seems "obvious" to me, since any structure which satisfies the Peano axioms will have natural number arithmetic. My formalist tendencies are at work here; the notion of an intended model seems somewhat foreign to me. There are many, many ways to construct the reals within ZFC; if they all act like reals, then what is the "correct" model? All statements true in a specifc model, but not in others are not part of real analysis; the "correct interpretation" is one where real numbers are taken as sui generis objects.

Finally, a change of topic. There seems to be a deep relationship between structuralism and phenomenology, which seems under-explored. Levinas, for one, makes a big deal of "existence without existents"; that is, being without thing-ness. This is exactly the idea of structuralism: we are studying mathematical notions without reference to a specific object to which the notion applies.

With You

I originally wrote this in Ghent (or Brussels?) in early January, but I've had trouble tying it all together. I'm still not competely satisfied with it, but I am content, so here goes.

Edit: Ugh. There are a few lines which just won't work... the first stanza is so seperate from the rest of the poem; I am sorely tempted to cut it out entirely, but I have very difficult-to-explain reasons for wanting to keep it.
Edit again: Yeah. I think the first stanza is the biggest problem. Until/unless I find a way to express what the first stanza is supposed to express, consider the poem to start with "Through days and day-dreams".
Edit again (again): I've removed the first stanza from this post.
****

Through days and day-dreams, I walk alone,
along ancient ways on cobblestones.

Yet in all my wandering, there is one thought
which fills my heart:
I would like,at last,
to be warm at Home.
In Bed.

A few thoughts on the officiating in the 2010 World Cup...

Edit: Sorry about the lack of linkage... I'm too lazy to dig back up all my sources

I'll try not to bore you too much with sports, but I'm super-stoked for the World Cup, and as always, I have some skepticism about some of the officiating.

I was excited last week when the officials were completely, always right. There were a number of times (in each game) when I said "are you kidding?" and then when watching the replay, I realized that the referee had made the correct call, and in a hard-to-see play. My hat's off to them.

Unfortunately, this hasn't kept up. A few days ago (I've forgotten which game), a goal was scored by a clearly off-sides striker. Why wasn't it called? The assistant referee was 3 or 4 yards up-field from the last defender. This is possibly pardonable in a club match, but not at the international level; especially in the World Cup finals.

Then today, we see an inconsistent, booking-happy referee in the Serbia vs Germany game. I'm really skeptical about most of the cautions he made. Regardless, he was inconsistent the whole match, and clearly was approaching his role with a very different mentality in the second half.

And of course, it'll be a while until people shut up about the USA-Slovenia draw... From the first moment I was questioning the referee-- he was inconsistent, missed some clear infringements, and called some spurious "fouls" where there was little contact, and a clean challenge. There were a number of nearly identical little pushes from behind throughout the match. One earned a caution, 2 (I think) earned the proper free kick they deserve, and at least 2 earned an absent-minded turn of the head.

The first booking had me laughing. Findley's yellow for a "handball" (clearly unintentional and off of a hand in a natural position-- viz, clearly not an infringement according to the Laws of the Game) had me groaning. The third booking was fair. The late-game caution near midfield was ever-so-slightly iffy. Had the referee not lost the benefit of the doubt with sub-par decision, I would hardly even bring it up, but his performance all game was questionable enough that I don't feel bad questioning it.

The booking on Josy's break-away was hardly cautionable-- the only reason I see for a booking was stopping a "clear goal-scoring opportunity", so if there's a booking, the guilty player should be sent off.

Finally, we get to the decision that Americans will be moaning about for months to come-- the spurious infringement during Donovan's (would-be goal-scoring) free-kick. Firstly: the announcers were wrong, it was not off-sides, the referee explicitly said it was for a foul. He did, however refuse to say what sort of foul, and who committed it.* Pictures of the play show what appears to be no less than 4 Slovenian defenders fouling American players, and I've seen only one picture which shows what may be an American foul-- Bocanegra appears to have his arms around a defender (Pecnik). In that picture, it looks to me like Bocanegra is falling, and (instinctively) lifting his arms to grab something. Watching the replay only confirms that pictures can never capture what's going on before set-pieces-- a pair of the apparent Slovenian infringements clearly were not, and I can only find 2 certain infringements: Radosavljevic bear-hugging Bradley and Cesar holding back DeMerit. Also, Bocanegra (as I suspected) is laid out at the beginning of the play (He's the guy rolling around when Edu shoots). I can't tell how legitimate his fall is, but considering Pecnik doesn't come down with him, he is clearly not holding Pecnik.

In any case, no less than 2 fouls in the action of the play are missed by the referee, and we have no idea what it was he did see.

*Two notes: first, it is officially (according to something I read on the internet...) a foul against Edu. Considering he hardly touches the only defender who's near him, the only possible call is offsides. Which the referee denied. And which he clearly wasn't.

Second, apparently this is coyness is within the referee's rights. Even in the post-game report, referees are not required to say (i)who committed an infringement, or (ii)what the infringement was, unless the player is booked.
During the game, fine, but in the post-game report? That is unacceptable, precisely for situations just like this. No one knows what the call was, and apparently, this includes the esteemed Mister Coulibaly. Fans need to know what it was that he saw.

Unlike many (American) fans, I won't say that Coulibaly should be investigated for any sort of gross misconduct-- he was hardly fair to the Slovenians, and this is not the first match where his decisions have been loudly questioned-- and I won't say that FIFA should rectify anything. I also (unlike many fans the world over) won't say that new technology should be used in-game to rectify wrong decisions-- part of the beauty of soccer is the pace, and I'd prefer FIFA avoid bad decisions by being more vigilant about using referees with a history of very controversial decisions (3 of the 5 Africa Cups he's refereed in!), than by slowing the game down. But I will say this:

• FIFA needs to force referees to say who committed an infringement, and what it was, at least in a post-game report-- allowing the referee to archive these after re-watching the game, so as not to slow down the game.
  
• FIFA needs to make public statements whenever a referee consistently makes controversial calls, and whenever a referee makes a highly controversial call. Either defending the referee by saying something along the lines of "You may not agree with his interpretation of the Laws, but we consider them acceptable", or admitting that the referee was wrong. A clear procedure for such situations (for petitioning for a statement, and for the subsequent review.)
  
• In the case that the referee is wrong, FIFA needs to administer punative actions. If a player acts in an unacceptable manner, they are typically fined, and often suspended for longer than the one-game suspension that comes with a send-off. Similar measures should be instituted for referees whose decisions are not within acceptable interpretation of the Laws.
  

Edit (About 20 minutes after posting): It seems FIFA is in fact reviewing Coulibaly's performance, and will likely be unassigned from future matches in the tournament. This is a good first step, but I'll be waiting for a statement from the tournament organizers... I'm also interested in hearing from FIFA regarding the censorship claims

Art and Meaning

A recent Abstruse Goose reminded me of Preface to Dorian Gray. Of course, people deride critics all the time. Especially modern art critics. I tend to defend art critics even though I don't understand them, because I know they are working from a different framework, a different background and dialogue than most people (including myself), and that this framework is not necessarily wrong.

But this Abstruse Goose reminded me of something that I don't like about... well, most people who discuss art, and this thing is what reminded me of Dorian Gray: Trying to find meaning in art is to murder it. The criticism and analysis of art should concern itself only with the aesthetics of the art-- the emotional complex that the art inspires. Yes, a piece of art can have a moral, or a meaning, but if the piece does not stand without that moral, then it is a failure as a work of art. The meaning should be extracted and the work discarded.

If something can be said, it can be said directly. "But," you say, "what about metaphors and paraboles?" Yes, they have their place, but they are not art, and should not be treated as such. Pilgrim's Regress is not a good piece because it tells the tale of someone coming to God; it is a good piece because it captures the beauty, the sehnsucht that helped to define Lewis's faith. Without capturing this beauty, it would be nothing. Speaker for the Dead is not a good piece because it questions the neo-colonialism inherent in anthropological methodology; it is a good piece because it captures so many emotional struggles.

I mentioned two novels, because novels most often do have a meaning; but the meaning is not what makes them good art. Aesthetic considerations are what make or break a novel. On the other hand, most visual art, and most good poetry do not have a meaning, and trying to suck meaning out a poem or picture is to fail to see the piece as art. This is unforgivable.

You may say that understanding the meaning helps to appreciate the aesthetics. Take my example of Pilgrim's Regress: is the aesthetic appeal not greater appreciated by understanding the God and the path to salvation? Yes, it surely is, but this understanding is assumed by the work, not expressed by the work. Someone approaching the work from a framework similar to Lewis's own will be better able to appreciate the work. But this says nothing about meaning. Understanding the background of a work, understanding the dialogue that the work is a part of and the "etymology" (if you will forgive the abuse of terminology) of the work will give someone a better appreciation for the aesthetics of the work. But to say that this is achieved by studying the meaning of the work is to commit cum hoc ergo propter hoc. Just because understanding the background helps both to understand the meaning and the aesthetics does not mean that understanding the meaning helps you to understand the aesthetics.

By the way, this is why you will rarely (never?) see me engage in a semantic analysis of a poem; I will study it syntactically, and I will study the way images are invoked, but I will not attempt to understand the author's intent, or understand the meaning of a poem. If the intent is anything but to create something of beauty, then I don't care.

Another poem.

Can't think of a title:

A man. A woman.
Sitting on the steps
of a Church in London.

A stifling silence.
Seconds fade. The man sighs,
"So what are we to do?
Will you be my tragic muse?"

A mirthless smile.
A soft sad kiss on the shoulder.
And a heavy, hesitant stance.

A man. Sitting
silently on the steps
of a churchyard, thinking.

Categories of paths; functors and natural transformations...

Somehow blogger screwed up, and a post I started (but never finished) a long time ago is "older" than another post of mine. It's about path categories, and higher categories... Hopefully worth a read?