"Concept Space"
This term is based on the same notion of "possibility space" as used in Swarm Intelligence. Possibility space is a mathematical term, though I don't presently have the book with me to verify whether or not Eberhart, Shi, Kennedy use it under the same definition. I also feel obliged to point out that it's been over a year since I read that book. Concept space is something I made up, which I will spend the rest of the post explaining.
Most people are familiar with the concept of space in general. By space, I do not refer to "outer space" (beyond the atmospheric limit), nor do I mean "open space" (the region of a place occupied by air and nothing else). Instead, I refer to the mathematical concept of space: topological space. (MathWorld)
The accuracy and usefulness of this construction is debatable: I find it a handy tool for explaining one aspect of my beliefs; admittedly, this has generally been restricted to myself. But it is no longer, which is why it's an entry here.
This aspect is the Definition.
Imagine, if you will, a Euclidean space, which is just a 3-dimensional region without boundaries. Say that every point in this space (definable by vectors with respect to axes) is a Concept. A concept is a discrete unit stored in your mind.
Now, we have a Term. A term is a spoken manifestation of a concept. Ideally, terms and concepts would be functions with bijection: term(x) = f(concept(x)) and concept(x) = f'(term(x)). This is, however, completely untrue in reality. Because every individual human being necessarily operates in a different context of reality than every other, their mapping is slightly different. The development of language, furthermore, would have been too taxing on the brain to have a one-to-one mapping: Theoretically, I'd say "googol" is too small a number for how many terms we'd need, even for one-month old child.
Thus enters the idea of regions. Instead of pinpointing a single concept, every term instead covers a range of concepts; in two-dimensions, this is a distance; in three-dimensions, this is a volume; in four or more dimensions, this is called content. When you utilize a term, it is the term that includes within its content the concept you describe.
So let's imagine our 3D space again. Let's say you have a sphere, and its name (i.e., term) is "bowl". Included within this space is every single bowl you have every experienced, from the term "Super Bowl" to "bowl you eat out of" to "a bowling game" (yes, bowl is a verb). Suppose you say "bowl" to someone who speaks English, but has never even heard of football (lucky them). They wouldn't include the Super Bowl in the various concepts underneath "bowl", for an obvious reason.
Now let's say we have another sphere. It's called "roll". Take a moment to imagine the many meanings this term has: a pastry, a revolving action, what you do after rocking, moving out, etc. This sphere has an intersection with the other sphere, bowl. That's because, in some cases, bowl and roll mean the same thing (okay, I'm stretching that a little, but they do). This is where synonyms come from.
But I was speaking of definitions. For that, we must consider a large number of people. They must all use the same word, and they all must have a reasonably similar meaning of the word. (And they might not!)
I have to bend to mathematics again, now. Imagine that you take the meaning of a word (we'll say "liberal") to be a sphere in the concept space with a radius of 5 units and a center at (3,2,5). (Please note that this makes a lot less sense in Euclidean space, but when you shift to something closer to infinity, the game changes.) That's a fair call. Now, we take person B. Person B takes the same word, but his sphere has a radius of 8 and a center point of (4,3,6).
Person A might think liberal is merely someone who votes the Democrat party line for no good reason besides simplicity and expedience. Person B, on the other hand, might think a liberal is a terrorist, and is someone who votes the Democrat party line... but because he's a terrorist.
If you draw out these spheres (get some graph paper, drop the last coordinate, and if it's easier, use squares instead; it doesn't matter), you'll find they 1) intersect, but 2) have some areas that are not shared. Person B's idea of "liberal" has a larger meaning than Person A's, but they both agree that a liberal is someone who votes along the Democrat party line. That's the intersection.
So we've got two people who have range of concepts they map from "liberal". Throw in another 200 million people (because we're America-centric tonight), and you have another 200 million spheres.
Yay.
Now we can get down to what a definition is, and whether or not it exists for a particular term. A dictionary entry, you see, follows this exact same process, only it's real, rather than my ideal "we know exactly what goes on in people's heads". Lexicographers look at how people use terms and provide such listings.
A definition is the intersection of ALL those spheres.
Ideally, there should only be one final definition, but this is not always the case. In this case, a term technically does not have a definition, but we pretend otherwise. And there is another case: where a definition becomes so small as to be useless. There are two instances of this: where a term refers to such a tiny region of the concept space anyways that it's actually useful, not useless (typically, proper nouns, chemical, and scientific names fall into this category); and where a term refers to several things too disparate to combine. An example of the latter is "black", which may refer to the color, an African American, a Friday, and a couple other things.
This is why names are so important, as I implied last night. The name is one of the ultimate devices of language: it pinpoints the concept so precisely that further description (refinement) becomes unnecessary. If I say "Michael Chui", you know who I'm talking about. (No, not the professor at the University of Indiana. Usually.) If I say "Firefox", you know. If I say "Microsoft", you know. If I say "America", you know. Such is the power of a name. Now, if I say "love", you don't know. If I say "happiness", you don't know. If I say "God", you don't know. These are terms that have failed to achieve the status of Name.
That about wraps it up, but I want to end on one final, longish note.
Multiculturalism, and why it's useful. Nevermind multilingualism; a language is just an excellent permeation of a culture. If you are rich, you should befriend someone who's poor. If you are from the West Coast, befriend someone on the East. A liberal should befriend a conservative. These are different cultures.
Culture, in terms of a concept space, is a set of axes. A different culture is a different set of axes. (And a language is to culture what a term is to a concept. Different cultures have different languages. Period.) If you know vectors, you know that you can define any point in a space with respect to a set of axes. "3 units down the X axis, 2 units up the Y axis, 5 units up the Z axis". Shift these axes and you have a different point with the same vectors. But why do we shift axes in mathematics?
Yes, that's right: a new perspective on the same thing. Oh, and you usually do it to make the problem easier to solve. Because it's a lot simpler from that angle.
Most people are familiar with the concept of space in general. By space, I do not refer to "outer space" (beyond the atmospheric limit), nor do I mean "open space" (the region of a place occupied by air and nothing else). Instead, I refer to the mathematical concept of space: topological space. (MathWorld)
The accuracy and usefulness of this construction is debatable: I find it a handy tool for explaining one aspect of my beliefs; admittedly, this has generally been restricted to myself. But it is no longer, which is why it's an entry here.
This aspect is the Definition.
Imagine, if you will, a Euclidean space, which is just a 3-dimensional region without boundaries. Say that every point in this space (definable by vectors with respect to axes) is a Concept. A concept is a discrete unit stored in your mind.
Now, we have a Term. A term is a spoken manifestation of a concept. Ideally, terms and concepts would be functions with bijection: term(x) = f(concept(x)) and concept(x) = f'(term(x)). This is, however, completely untrue in reality. Because every individual human being necessarily operates in a different context of reality than every other, their mapping is slightly different. The development of language, furthermore, would have been too taxing on the brain to have a one-to-one mapping: Theoretically, I'd say "googol" is too small a number for how many terms we'd need, even for one-month old child.
Thus enters the idea of regions. Instead of pinpointing a single concept, every term instead covers a range of concepts; in two-dimensions, this is a distance; in three-dimensions, this is a volume; in four or more dimensions, this is called content. When you utilize a term, it is the term that includes within its content the concept you describe.
So let's imagine our 3D space again. Let's say you have a sphere, and its name (i.e., term) is "bowl". Included within this space is every single bowl you have every experienced, from the term "Super Bowl" to "bowl you eat out of" to "a bowling game" (yes, bowl is a verb). Suppose you say "bowl" to someone who speaks English, but has never even heard of football (lucky them). They wouldn't include the Super Bowl in the various concepts underneath "bowl", for an obvious reason.
Now let's say we have another sphere. It's called "roll". Take a moment to imagine the many meanings this term has: a pastry, a revolving action, what you do after rocking, moving out, etc. This sphere has an intersection with the other sphere, bowl. That's because, in some cases, bowl and roll mean the same thing (okay, I'm stretching that a little, but they do). This is where synonyms come from.
But I was speaking of definitions. For that, we must consider a large number of people. They must all use the same word, and they all must have a reasonably similar meaning of the word. (And they might not!)
I have to bend to mathematics again, now. Imagine that you take the meaning of a word (we'll say "liberal") to be a sphere in the concept space with a radius of 5 units and a center at (3,2,5). (Please note that this makes a lot less sense in Euclidean space, but when you shift to something closer to infinity, the game changes.) That's a fair call. Now, we take person B. Person B takes the same word, but his sphere has a radius of 8 and a center point of (4,3,6).
Person A might think liberal is merely someone who votes the Democrat party line for no good reason besides simplicity and expedience. Person B, on the other hand, might think a liberal is a terrorist, and is someone who votes the Democrat party line... but because he's a terrorist.
If you draw out these spheres (get some graph paper, drop the last coordinate, and if it's easier, use squares instead; it doesn't matter), you'll find they 1) intersect, but 2) have some areas that are not shared. Person B's idea of "liberal" has a larger meaning than Person A's, but they both agree that a liberal is someone who votes along the Democrat party line. That's the intersection.
So we've got two people who have range of concepts they map from "liberal". Throw in another 200 million people (because we're America-centric tonight), and you have another 200 million spheres.
Yay.
Now we can get down to what a definition is, and whether or not it exists for a particular term. A dictionary entry, you see, follows this exact same process, only it's real, rather than my ideal "we know exactly what goes on in people's heads". Lexicographers look at how people use terms and provide such listings.
A definition is the intersection of ALL those spheres.
Ideally, there should only be one final definition, but this is not always the case. In this case, a term technically does not have a definition, but we pretend otherwise. And there is another case: where a definition becomes so small as to be useless. There are two instances of this: where a term refers to such a tiny region of the concept space anyways that it's actually useful, not useless (typically, proper nouns, chemical, and scientific names fall into this category); and where a term refers to several things too disparate to combine. An example of the latter is "black", which may refer to the color, an African American, a Friday, and a couple other things.
This is why names are so important, as I implied last night. The name is one of the ultimate devices of language: it pinpoints the concept so precisely that further description (refinement) becomes unnecessary. If I say "Michael Chui", you know who I'm talking about. (No, not the professor at the University of Indiana. Usually.) If I say "Firefox", you know. If I say "Microsoft", you know. If I say "America", you know. Such is the power of a name. Now, if I say "love", you don't know. If I say "happiness", you don't know. If I say "God", you don't know. These are terms that have failed to achieve the status of Name.
That about wraps it up, but I want to end on one final, longish note.
Multiculturalism, and why it's useful. Nevermind multilingualism; a language is just an excellent permeation of a culture. If you are rich, you should befriend someone who's poor. If you are from the West Coast, befriend someone on the East. A liberal should befriend a conservative. These are different cultures.
Culture, in terms of a concept space, is a set of axes. A different culture is a different set of axes. (And a language is to culture what a term is to a concept. Different cultures have different languages. Period.) If you know vectors, you know that you can define any point in a space with respect to a set of axes. "3 units down the X axis, 2 units up the Y axis, 5 units up the Z axis". Shift these axes and you have a different point with the same vectors. But why do we shift axes in mathematics?
Yes, that's right: a new perspective on the same thing. Oh, and you usually do it to make the problem easier to solve. Because it's a lot simpler from that angle.