As my last post might have illustrated, it actually takes a fair amount of words to spell out an axiom; indeed, axioms are arguably arrived at by talking, writing, or thinking a lot, which is exactly what Axiomatix is all about. Eventually the whole process can be boiled down to some essential slogan or saying, simple and poetic enough to be memorable, one that can then be explicitly applied to a situation, not as a universal law but instead as a rule of thumb. Usually such an axiom has to be reconstituted - "just add water" - or at least recontextualized so that people can understand it; in other words, it is actually the relations of that axiom with situations or with other axioms that allow it to make sense.
This leads me to another axiom I'd like to expand on here, one that makes further sense of Two Descriptions Are Better Than One: Relation is the Smallest Unit. This axiom is a saying that Donna Haraway likes to use, and one that took me some time to understand. Indeed, what I like best about this axiom is the way that it turns common sense on its head, giving logical priority to difference rather than sameness. Rather than consider that "relation" is something that happens between pre-existing "units", Relation is the Smallest Unit suggests that it is in fact the relating that is the most basic and fundamental "unit". Consider what this axiom does, for example, to Platonic idealism: while Plato argued that the very existence of chairs or triangles implied a universal idea of chairNess or triangleNess, Relation is the Smallest Unit suggests instead that chairs and triangles are not only constituted by their internal relations but also by their external ones. You don’t get a chair without certain relations between the materials that make it up, without its relations with other kinds of furniture and other objects, or without its relation to some critter – human or otherwise – who might sit on it, knock it down, or what have you. Platonic chairNess, then, can be seen instead as a vastly complex knot of relations which for convenience we call “chair”.
Applying this axiom in our practices and theories can sometimes be hard, because the folk philosophy we grow up in and speak inside of in our everyday lives leads us instead to see a simplified world of discrete and separate objects and ideas which can then be related to one another. Taking it seriously, we can understand – for example – that “defining” something by what it “is” is hardly effective, and that the best way of defining things is through their relations with other things.
We can also understand that thinking never actually happens in just one head – or, to rephrase it as an inconsistent, ornery, and downright paradoxical axiom: It Takes Two to Cogito. Comments? Rants? Axioms?
Shall we cogito?
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I started linking this idea to category theory but realized I was writing an introduction to category theory, which I don't want to do here. Objects--and in fact structures which are so fundamental they appear in different areas of math--are frequently distinguished uniquely in terms of that which maps into and maps out of them. If one can throw away one object and in its place leave maps into and out of neighboring objects, then one can do this to the neighbors as well. Eventually all that is left are mappings. To make this precise, one probably still wants a notion of object, but, for example, an element x of a set X may be instead considered to be a map (say, f) from 1 to X (the map which takes 1 to x). Then a map (say, g) from X to Y, which is defined by its behavior on elements in X, may alternatively be defined by how it follows maps such as f. I.e., g(f(1))=g(x), so g is determined by the way it acts on mappings (such as f). The point is that mappings can hold themselves together much like ice bricks in an igloo do, that by having neighbors they are related to, they no longer need to be cemented to anything!
ReplyDeleteI could go on and on, but I won't out of sense.
I'm interested in your 'it takes two' phrase. I don't agree in the strictest sense, but I see that we don't get too far beyond each other in thought. You cannot think a lot about something really novel until you make yourself a path of definitions (i.e., relations) into that area. I don't see how thinking only happens in one head, though, perhaps more explanation is needed.
Thanks Andrew. If I'm understanding you correctly, though, category theory is a kind of metamapping, by which I mean that the map maps other maps, which thereby need to be redefined as territory, yes? So there's still a territory.
ReplyDeleteRe: It takes two to cogito, I guess I'll go into that later, but the point is that thinking only happens in more than one head, not in one. Ergo the paradox: "it takes two to 'i think'". Briefly, the very tools of thinking are shared between people, so how could thinking happen if there were only one?
On another note, I wish we'd played Go more often. I watched this Guy Ritchie film that I highly recommend, "Revolver". It's not a great film, but it's about the art of the con and the art of the game, and one of the points is that you can only get smarter by playing someone smarter than you. I could have gotten a lot better at Go if I'd been willing to be totally crushed by you, but my ego wouldn't let me. See the movie, you'll understand. I think you can watch it on TVShack.
cheers,
Sha
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ReplyDeleteKGS is a great go server/community. I used to care about winning, now I'm addicted to playing online and I play to kill time and relax, frequently lazily and unbothered by loss. You should give it a try Sha. My KGS name is context.
ReplyDeleteYes, you're right about the mappings becoming territory. You can somewhat substitute relations in for object, but then the sets of relations become the objects, in some sense. I don't have anything else to say about CT. It's mostly difficult to describe without drawing examples from lots of mathematics. Even a blackboard and a few drawings can convey the idea of a universal mapping property, but it's not so important here.
I have immediate problems with "thinking cannot happen in one head", imagining someone who writes novels and burns them before anyone can read them. However, I am convinced our thought, what we recognize as thought, is founded on communication, so that a "god" (or any being that has no need to communicate) existing by itself could not think in any way recognizable to us as thought.
Another axiom-like slogan i like: "there are no excuses." I don't mean it in the usual, pulling oneself up by bootstraps sense, but somewhat similarly. It's a rephrasing of "it is what it is." Of course there is a place for excuses, but generally, since there is always an explanation for why things didn't turn out otherwise, it's healthier to accept fate and effect it (or just affect it) than to be its victim. I know it sounds like something a junior high school guidance counselor would say, but still it's valuable to me.
Love this: "Relation is the smallest unit" - I'm stealing it as a defense against reductionism. As an axiom it's really a very elegant expression of a phenomenological and experiential world-relation rather than a corpuscular one. Much appreciated and reminds me very much of Krishnamurti that I've read (Freedom from the Known).
ReplyDeleteAlso love Go. I play on IGS and KGS too much, probably more than I should given what else I should be doing, and I should be better by now having played for 20 years- woefully only 10k on IGS, 4k KGS
ReplyDeleteAlso Sha your conviction that thinking is communal is buttressed by much thinking and writing in sociology, psychology and lit crit - check out Bakhtin's notion of the dialogic self. If language is the medium of thought, and language (our particular language, not just Chomskian mappings of it) then we think in our "mother tongue", which must be taught to us in relation to others (or the m/Other).
ReplyDeleteWell, I was once rated at 13k (proceeding then to kick the ass of someone rated at 5k), so what to do with these ratings? Anyway, thanks for playing Ersatz Bling, please friend me on FB, and I am totally down with Bakhtin without however having ever bothered to read much of his work. (I have similar relationships with, say, Derrida and Lacan.) Indeed, the whole transitive aspect of theoretical and philosophical thinking is living proof of "It Takes Two to Cogito": when my own mystra - Donna Haraway in the current case - has ingested, digested, and excreted a gazillion-and-one thinkers, then do I really need to do the same? As Wittgenstein once put it, there is always a last house on the street - but you can always build another one.
ReplyDeleteremain in light,
Sha
I don't get it, still. "Thinking *is* communal" is a surprise to me, since I believe some thoughts belong to the individual. I can see it is meant to be a surprise, and if it's an exaggeration it is meant to convey some truth about how thinking is socially formed. I'm sure thinking is socially formed, to some great extent, but I also know the individual is stuck with himself and his previous selves in a way he is not stuck with others, and is privy to experiences through those past selves in a way that he is not privy to others' experiences. That's very 'common' sense, of course. I just don't see any reason to shed it.
ReplyDeleteunit - union thing - relation think.. I am a chair if you sit on me. I think we are more hive minded than we realize. "Our" thoughts showing up in the mental nets we weave with the patterns of knowledge obtained from others. The thoughts that appear in my head are a result of the way my net is woven. Though I do like to think that we each have our own unique design, like snowflakes.
ReplyDelete