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Saturday, January 13, 2007day link 

 Dimensions of Comprehension
picture I've long had the sneaking suspicion that the solution to a great many human problems would be found in the understanding of dimensions. We're just operating in too few dimension most of the time. Or, said a different way, we live in an increasingly multi-dimensional world, but we only have tools and thinking processes for dealing efficiently with one, two or three dimensions.

So, we're often like flatlanders. We operate in a limited number of dimensions and when something with a higher dimension comes along, we just don't get it. It is incomprehensible. But sometimes, if one can manage to step up one dimension, what before was mysterious, super-natural and confusing suddenly becomes clear. From higher dimensions , the lower dimension are easy to understand.

I've written about it here, here, here, here, here, here or here.

I was reading Tony Judge's summary and reflections on a book by Ron Atkin called "Multidimensional Man; can man live in 3-dimensional space?". I don't entirely get it, but in brief it is about the problems of comprehension in different dimensions within organisations. 0-dimensional comprehension would be if one can only understand separate, isolated issues. One point or another. 1-dimensional would be that one can comprehend the range between such points. Like, political left or right, or anything in-between, but nothing else. And with 2-dimensional, one could naturally comprehend several things at the same time and the various nuances between them. For example, one might have more or less economic control by governments, and more or less moral authoritarianism, and one might understand both at the same time, and the various ranges of possibilities, even though they go in different directions.

But let me re-think it in my own way here, as that is easier.

We're used to thinking that dimensions are physical. You know, we live in a 3-dimensional world, where things have length and breadth and height. And maybe we could think of time as a 4th dimension, although we only half understand it. Objects and events have a dimension in time, but we don't think of it that way. We vaguely refer to the state of something earlier, or the possible state of it later, but we have a hard time thinking of it as one object in 4 dimensions. And we're sort of stuck on believing that time moves only in one direction, and that the future kind of doesn't exist. So, time is at best a half dimension for us.

But all of that is just one way of looking at dimensions. More generally, dimension is the phenomenon of degrees of freedom. So, it isn't just about standing in a room and trying to imagine how one might step off in a right angle to the three dimensions one sees, which one logically should be able to do if there were four dimensions. That's interesting enough in itself, but dimensions are at play in different ways.

I'm talking about degrees of freedom in the sense of independently moving variables. The number of coordinates needed to specify something. The length, breadth, height thing is a special case of that. If you're talking about the position of an object in a room in relation to you, it can theoretically be moved left or right, forwards or backwards, up or down. So, we need an X, a Y and a Z to describe its position. I might argue that they aren't really independent of each other, but traditionally we use 3 variables, or 3 dimensions to describe it. Anyway, that only described the position. I'd have to introduce a number of more variables to describe what it is I put in that position. Like, what is this thing's shape? Its color, density, weight, composition, etc. Even just the color, we usually use 3 dimensions to describe that, and it probably doesn't have only one. Let's say the object was a ball. That's pretty simple, but would still require a whole bunch of variables to describe even approximately.

Still, I can look at a ball lying on the floor, instantly comprehending what I'm seeing, and some of the implications of that. Despite that an adequate description of what I'm seeing would require variables in a whole bunch of dimensions.

The thing is that they aren't moving independently from each other at the moment. For that matter, they aren't moving at all, as the ball is just lying there for the moment. So, I can comfortably perceive the ball as "one thing". In thinking about it, I can even reduce it just one or zero dimensions in order to keep track of it. It is just one point, "the ball", and I can think quite logically about it based on that, without having to worry about balls in other places, or balls of entirely different dimensions or anything like that. Doesn't matter, because they don't have any bearing on what I do with this ball.

So, in terms of my own comprehension, my own mental world, I can regard the ball on the floor in 0 dimensions. I can then just as easily make 1-dimensional decisions about it. Should it be on the floor or up on the shelf? This point or that? The ball is unvariable, the shelf is unvariable. I can put it in various places, but the thought process really only involves one dimension.

If my daughter who put the ball on the floor has a different plan with it than I do, I might have to deal with 2 dimensions. My idea of where to put it, and her idea of what to do with it. Two variables quite likely moving independently of each other.

If there were more kids involved, it undeniably gets more complicated, but it might still be a two-dimensional problem, unless there were other issues involved. Say there were two kids, and one of them would start crying if she didn't get the ball, and somebody else might think I had been mean to her. Then we have maybe 3 dimensions.

You can argue about the exact number of dimensions, of course, but you get the point. How many pieces or dynamics are operating at the same time, independently of each other.

You know the often-quoted concept that a human being can keep track of at the most 5-7 different things at the same time. You can comprehend and act logically in regards to 5 to 7 independent variables at the same time, at the most, but after that you start getting confused. Your picture of what goes on is no longer coherent. You'll start dropping balls if there are too many things moving in too many directions, independently of each other, at the same time.

This has nothing to do with the number of pieces involved, or the number of people, or the number of variables in terms of physical characteristics. You can keep track of 200 cars in a parking lot, even though they have different colors, and they each consists of thousands of pieces. That has nothing to do with it, as long as you, as the parking lot attendent, can operate with a very limited number of independent variables at the same time.

So, there's this concept of dimensions that has very little to do with how many dimensions there are in our physical space. It is a different kind of space that instead is based on the degrees of freedom of the stuff we feel we have to deal with.

We have to deal with more dimensions than we used to. The internet and other communication technologies are in part to blame for that. Somebody might call me on my phone while I'm walking on the street. That adds some variables that didn't exist before. I might have to deal with something I otherwise wouldn't have had to deal with. And since I live on the Internet part of the time, I participate in various kinds of activities I couldn't have participated in before, most of which are moving independently of each other in different directions.

And information adds all sorts of dimensions. A few hundred years ago I might have spent my days hard at work in a field, and no information was very important, other than a bit of knowledge about my work, and knowing who my friends and enemies were, and none of that was abstract. But today we're immersed in abstract relations based on information.

Economics adds dimensions too. However many dimensions I consider my house to have, several more are added for me to consider that I need to pay the rent, and where the money for that comes from.

Intelligence is in part applied to predict the consequences of different types of actions, and choosing the proper action to take at a given moment. But the more variables there are, particularly when it is more than you can comprehend, you might become a bit stupid and no longer able to make optimum decisions.

What happens if you continously are faced with 10-dimensional problems, but you're only able to comprehend 5? You'll use those to the best of your ability, of course, but you'll be lacking something.

Either you have to evolve, to think in more dimensions, or you have to get tools that do some of it for you. In principle computers can deal with problems with many dimensions as easily as problems with few dimensions. OK, not quite true, it gets harder the more dimensions there are, but a computer doesn't get confused the same way you do.

Say you need to find the best time for a meeting with 10 other people. That's very hard to do if you had to call them one by one. A computer could do it easily, if their respective schedules and priorities and criteria were stored, accessible and structured well. No sweat, in principle. But my calendar program is really dumb. I can put events into it, set times, and move or delete the events if plans change. But it doesn't help me at all to organize my time. I'd like it to.

Anyway, the point I'm trying to get to is that we're living in a world of multiple dimensions that have relatively little to do with the physical dimensions. It makes our world much more complicated than it would be if we were just talking about physical dimensions. We have good tools for navigating in physical dimensions, like maps and compass directions and GPS and measures. We don't really have very much trouble with that. What makes life complicated is that most of what's important has nothing to do with clear coordinates like that. We don't have very good tools for navigating all the other dimensions. Oh, we have tools for various pieces of it. I can make a budget to navigate the dimensions of paying my rent. I can have an e-mail program that keeps track of my conversations. But there are no general tools for visualizing all the dimensions in your life.

It is in many ways an invisible problem. People will express their opinions and knowledge about all sorts of things, without understanding how many dimensions there are to what they're talking about. Politicians, or even scientists, will reduce the situation at hand to a number of dimensions they have a suitable model for, without regard to the fact that there usually are more variables involved.

Forget about even 5-7 independent variables. Most public discourse limits itself to 1 or 2. If you write about an article about some subject you're knowledgable in, your readers would lose you if you talk about more than 1 or 2 dimensions of it at the same time. You can say that there's this one factor A and another factor B, and you can describe the hypothetical situation quite clearly, but if you introduce a bunch more, people are just going to think that you don't know what you're talking about. But what if reality truly is based on a whole bunch more dimensions than you easily can talk about?

And what if our world has evolved into more dimensions without us having noticed it?

Nothing dramatically changed about the physical dimensions. The diameter of the planet is roughly the same, and the distance between New York and Berlin didn't change much, and you still need length x width x height to choose a bookcase.

But your life has more dimensions to it than it did 20 years ago. And I'm afraid that your brain, like mine, probably is a couple of dimensions too short to really make sense of it.
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