In its standard form, conceptual analysis assumes the “classical view” of concepts, that a “concept C has definitional structure in that it is composed of simpler concepts that express necessary and sufficient conditions for falling under C.”
I suspect that I am a bit slow on the uptake here, but I’m not sure what’s not true. (Something was thought and now we know it isn’t so?)
On the one hand, I understand that a set might be defined as a collections of objects satisfying just a handful of necessary and sufficient conditions, and that humans often think about simple sets that are defined in this way. For example, the set of primes is particularly easy to define with just a couple conditions.
It seems a bit awkward at first, but I suppose I could also think of any human-defined set as a ‘concept’ for the person considering the set. So for example we have a ‘concept’ of prime number.
What about the other direction? Do all human concepts map to sets? In which case, are these sets explicitly defined in our brains by a set of necessary and sufficient conditions?
I would have guessed the answer should be “yes” for any concept that we’re certain actually maps to a set (like the set of ‘fish’). Even if we do have fuzzy boundaries for a category; this fuzziness could reflect our uncertainty about which set we’re pointing to or uncertainty about the rules that make the set. (Doubtless I’m echoing some 15 year stale argument...) Explicitly, uncertainty whether an olive is a fruit could reflect either one of the following problems:
The person isn’t certain which set called “fruit” you mean. Do you mean fruit1 defined as all plant organs that have seeds, or fruit2 defined as plant organs that are exceptionally sweet?
The person isn’t sure what the conditions are for being in the set “fruit” and supposes that olives could belong to the set with some probability. That is, the uncertainty reflects their knowledge about the set rather than the fuzziness of the set itself.
Are we discussing whether a set maps to something ‘real’ in a person’s brain? It seems natural to think of objects belonging to or not belonging to a set based on criteria—is this not what my brain does when it thinks about a set like ‘fish’? Does my brain do this sometimes? Or is the question whether my brain does this always?
Finally, is it possible to have a concept without defining a set? … Perhaps if a concept is just pattern matching? For example, let’s suppose I see the sequence 2, 4, 2, 4. I have a ‘concept’ of a pattern that it is alternating 2s and 4s. Have I necessarily defined a set (for example, the set of patterns ABABAB...) to predict the number that comes next?
In summary, I would like to understand more clearly the conclusion regarding what cognitive science has learned about the relationship between sets and concepts.
Even if we do have fuzzy boundaries for a category; this fuzziness could reflect our uncertainty about which set we’re pointing to or uncertainty about the rules that make the set.
But there aren’t any rules out there in the universe that define concepts; the concepts are in our heads. You could certainly argue for picking a strict definition (i.e. figure out exactly where a “small pile of sand” becomes a “heap of sand” using T-Rex’s method), and convince everyone that it’s the best definition… but that’s just a different (though possibly better) concept, not a more accurate representation of a set rule in either sense of the word “set”.
Focusing on the boundary between member and non-member is not typically a useful practice anyways (though it can be fun, or meta-useful as when discussing cog-sci itself). As pointed out in the OP, the parts of a concept people use the most are the ones that contain the most obvious members of the set. If you’re squirreling around in the fuzzy border-lands of “is it a heap? is it not?”, then you need to go back and figure out what question you’re actually trying to ask, instead of arguing about a concept that doesn’t have much immediate usefulness.
The set of ‘heaps’ is an interesting example of a set.
Do you suppose, given our sense of pleasant dissonance with respect to trying to identify the smallest number of elements in a heap, that one of the necessary qualities of a ‘perfect’ (most typical) heap is that a person doesn’t know how many elements there are?
Or, rather … thinking about it further—being a ‘heap’ refers to how it appears to have been formed, with items randomly piling on top of one another. And at first, it is difficult to imagine ‘heaping’ of just one thing. Yet I suppose a person could heap themselves on the floor, and then given that context, it suddenly makes sense for there to be a heap of even one thing. Say, a pair of pants thrown carelessly on a chair.
So just a few moments were needed to trace the correct conditions for being a heap, and then it makes sense. The set of ‘heaps’ is defined based on the verb (items belong to the set based on the believability that a configuration may have been formed in such a way) and not on the number of elements.
Hm, that makes sense. For a similar fuzzy-quantity-border problem to what I was thinking of originally, though, consider these other nouns:
Crowd
Bunch
Drop (of liquid)
Bundle
Shard
And so on. And that’s not even getting into adjectives where the fuzzy-border property is even more pronounced (tall, short, big, small, heavy, light).
Some of these words I have ‘concepts’ for, others I don’t. If I don’t have a concept for the word, it seems to be understood just at the association stage—I can only come up with a list of contexts where I would use the word.
If a set is defined by necessary and sufficient conditions, then having an association means that you have some sufficient conditions but you don’t know the necessary conditions. That is, you can identify a region of ideas that are in the set but you don’t know where to draw any boundaries.
At this sufficient but unknown-necessary-conditions stage, you don’t really understand what the word means (you don’t have a well-defined concept) because you need to be able to delineate something that is not in a set in order to really understand anything about what a set is. I agree that not knowing the exact boundary may not be important—but it is important that there is some known interior and some known exterior. My conclusion is that any concept that is understood at any elementary level must have both sufficient and necessary conditions.
Regarding those words which are more associations than concepts, on NPR today the following sentence stuck out as relevant:
‘headwaters is kind of a political word to a lot of people around here; nobody really knows what that means or wants to agree that is where the mine is but—’
Hearing this statement sort of cemented a hypothesis I was considering (in a draft reply to this comment this morning) that many words don’t have real concepts behind them because their main use is to signal something. For example, I would rarely use the word ‘crowd’ when thinking to myself. I would only use the word ‘crowd’ hyperbolically, to indicate to someone that they should imagine more people than they would otherwise expect. In other words, the word augments a concept of how many and the meaning depends on a particular context of myself and a listener.
I think all words have some sort of concept behind them (for example, ‘headwaters’) but since they’re not literally used to convey that concept very often, a speaker of that language can be confused about or forget the concept behind the word.
In an internal dialogue, I would only use the word ‘crowd’ if there was literally jostling of elbows; I think that is the “original” concept for me. What does ‘crowd’ mean to you?
Hm, I see where you’re coming from, and I agree with you so far as it goes when mapping between sets and concepts. But to switch back to talking about words instead of concepts, I am not sure why you (seem to be) going with a set-based approach. Thinking of words as fuzzy regions seems more useful to me when trying to analyze how words are used, and when trying to figure out how words ought to be used.
As shown in the OP, people tend to use words not as strict sets but as regions in thing-space with more and less typical members (i.e. a bluebird is a more typical member of “birds” than a penguin or an ostrich). Though some situations will demand strict borders, in general the fuzzy region approach seems more beneficial, because it allows for quick low-level inductive reasoning. If you know that a bird is typical in bird aspects A and B (say, it has feathers, and it has a large wingspan), then you can more confidently predict that it’s typical in bird aspect C (it can fly). It’s less brittle to use probabilities in situations like these than strict boolean sufficient-or-necessary values.
Though some situations will demand strict borders, in general the fuzzy region approach seems more beneficial, because it allows for quick low-level inductive reasoning
You’ve convinced me on the importance of fuzzy sets. I’m sure quick low-level inductive reasoning is of vital importance, but even more immediately it seems that flexibility in meaning is useful for communication. I don’t have to hunt for the exact right word, many approximate words will do, and I can describe new concepts by stretching an old one.
However, when you write:
It’s less brittle to use probabilities in situations like these than strict boolean sufficient-or-necessary values,
I don’t see how sufficient-or-necessary defined sets don’t allow fuzziness. For example, saying that a bird must have feathers (a necessary condition) and that a bluebird is a bird (sufficient condition) then there is still plenty of gray area for penguins and ostriches.
I think of sufficient conditions as being associations. We learn that penguins, ostriches and doves are all birds so those are sufficient conditions. A necessary conditions puts an edge on your set—staplers aren’t birds because they don’t have feathers.
On the other hand, a stapler that looks like a bird could be thought of as a bird to oneself. If anything has enough of the characteristics of a bird you could consider it one. For example a small plastic object with no feathers whatsoever is a ‘birdie’ because it moves like a bird.
So now I’m leaning towards agreeing. We just have associations, things are like other things if the characteristics overlap, and there aren’t any necessary conditions for deciding if something is ‘like’ something else.
[...]it seems that flexibility in meaning is useful for communication. I don’t have to hunt for the exact right word, many approximate words will do, and I can describe new concepts by stretching an old one.
Yes, strongly agreed. This idea makes me want to think of adjectives as tugging a concept-defining region of thing-space in a new direction.
That works especially well in languages like Lojban, where adjectives and nouns are not distinguished from each other. For example there is “blanu” which means “x is blue” and there is “zdani’ which means “x is a house”, and you can say “blanu zdani” (blueish-house) just as easily as “zdani blanu” (houseish-blue).
[...]saying that a bird must have feathers (a necessary condition)
That actually is a good example of a brittle requirement, in ways that are even more directly problematic than shuttlecocks and bird-shaped staplers. What about a plucked chicken? What about a duck that, due to a genetic disease, never had feathers? What about (NOTE: This example isn’t valid re: real evolutionary history) a member of an intermediary species between pterodactyl and modern birds?
Not that it particularly affects your point, but pterodactyls are not genetic precursors to birds (they split off before the clade Dinosauria,) and feathers predate the first true dinosaurs capable of flight.
That actually is a good example of a brittle requirement,
Yeah, good point. I’m entirely convinced. Even for an apparently straight-forward category like ‘bird’, there’s not a single necessary condition you can point to. Even if there are some examples of categories with necessary conditions (I don’t know), this is evidence that the necessary conditions aren’t an intrinsic part of the way we structure a concept.
I don’t think beauty is a ‘good’ concept in every day usage. Perhaps this is because it doesn’t map to a set for me!
On the other hand, in a context where the word ‘beauty’ is being used in a specific way that I could get a handle on, I can imagine knowing for that context what concept ‘beauty’ is pointing to and being able to identify the set.
Note this doesn’t have anything to do with beauty being subjective or a 2-place word. If I know what concept to tie ‘beauty’ to, I could then know what set of things are beautiful. Note also this would be a very specific set (say, ‘Beauty2035JL’) since there would probably be many similar sets that I could immediately encounter.
Then, if I’m not mistaken, the point being made here is that our idea of beauty is not a single, specific concept, but a vague feeling that could correspond to any of myriad different concepts. Therefore, we can’t give necessary and sufficient conditions for a thing to be beautiful—unless, as you mention, we determine which concept of beauty we’re referring to.
(It sounds to me like “concept” means one thing when lukeprog says it and something else when you say it.)
Oh, my point was that (for me) ‘beauty’ isn’t a concept but just a token that gets bandied about. I have some faith that there is a concept behind the token when other people use the word, and I have used the word in contexts where I’ve estimated that the concept is somehow ‘close enough’ to what might be meant by beauty … but simultaneously I’m aware I’m not communicating any specific concept and probably not communicating the one I intend.
Even a vague feeling could point to a concept. If you have a vague feeling about ‘beauty’ that could correspond to any of a myriad of different concepts, then that is probably a set. (The set of all things you do tie or would tie to that feeling at that moment.)
I’m liking the idea that any concept must point to an actual set, though, and I think I’ll go with that definition in my thoughts from now on. So: a ‘concept’ is something in my brain that defines a set.
… But now it is time to go back to lukeprog’s post and check what he meant by concept and start using that token with his meaning in this context..
Even a vague feeling could point to a concept. If you have a vague feeling about ‘beauty’ that could correspond to any of a myriad of different concepts, then that is probably a set.
Or it’s any of myriad different sets.
But yes, trying to standardize our terminology will probably be helpful!
a ‘concept’ is something in my brain that defines a set
I don’t think you do have anything in your brain that defines a set, except when you formulate a precise verbal or other description of an actual mathematical set. Which is not what you’re doing when you ponder a concept like beauty.
In the instant that you think about whether something is “beautiful” the concept is a transient configuration of neuron state that at one particular point in time associates something in your memory or perception with the word “beautiful”. In which case it’s not any more meaningful than any other random association that happens to occur.
How would you tell this is what is happening instead of your brain containing a well-defined set? First note, it is not going to be consistent—you will change your mind. It’s also not going to be complete—there will be things where you don’t have any particular feeling about whether they are beautiful. Also, you will be able to say whether things are beautiful even when they aren’t a discrete thing and have boundaries just as fuzzy as the concept you relate them to (e.g. a beach with surrounding landscape). And you will do this without even noticing.
In a way, I completely agree with you. I agree with you in the sense that I think you are slicing reality in exactly the right way, and any remaining disagreement is just definitional.
transient configuration of neuron state that at one particular point in time associates something in your memory or perception with the word “beautiful”. In which case it’s not any more meaningful than any other random association that happens to occur.
Agreed, we should distinguish this random-association-type thought from the brain process (however it may be done) of mentally defining a set.
What I want to do next is say that at this random-association stage of a thought process, you don’t have a concept. Ithink you haven’t seized a concept unless you’ve defined the set. The concept, if it is lurking there, hasn’t been understood and hasn’t been ‘owned’.
the concept is a transient configuration of neuron state
Yes, I think this transience is what makes it difficult to recognize and discuss concepts correctly. For example, in several places ‘concept families’ have been mentioned, as though the primary object is a fuzzy set and at instances of thought we’re picking out particular sets from this fuzzy family. I see this reversed: sets are always specific, but our thoughts transition so fluidly from one set to a nearby set that we imagine there is a single larger fuzzy set that these sets are coming from.
For example, when we think of the set of fish, we are likely to first consider something like ‘the set of animals that look just like Nemo, but with any color variation’. (This is what was referred to as a ‘most typical’ member.) Then a few seconds later we remember sharks are also fish and throw them in. Both sets are called ‘fish’ in our minds from one moment to the next, but they were different sets and our brain can distinguish them, we just didn’t bother to track the differences as our concept of ‘fish’ evolved. So a thought process will evolve a lineage of sets SetFish1-->SetFish2-->SetFish3 over the course of a few seconds.
I think the fact that our brain can easily distinguish them all (via necessary and sufficient conditions, if not actual single-word linguistic tokens) is evidence that we understand the individual sets first, and the understanding (concept) of a ‘fuzzy set family’ comes from the observation and generalization of this lineage.
Note: I mean, this is what I think now. I’d be interested in a different paradigm for how my brain understands a concept and/or a set.
I’m voting this comment down, not because I don’t think the link is relevant or useful (because it certainly is!), but because in this context linking to it without any explanation comes across as rather rude, and is less likely to be clicked on by people who would benefit from reading it.
I suspect that I am a bit slow on the uptake here, but I’m not sure what’s not true. (Something was thought and now we know it isn’t so?)
On the one hand, I understand that a set might be defined as a collections of objects satisfying just a handful of necessary and sufficient conditions, and that humans often think about simple sets that are defined in this way. For example, the set of primes is particularly easy to define with just a couple conditions.
It seems a bit awkward at first, but I suppose I could also think of any human-defined set as a ‘concept’ for the person considering the set. So for example we have a ‘concept’ of prime number.
What about the other direction? Do all human concepts map to sets? In which case, are these sets explicitly defined in our brains by a set of necessary and sufficient conditions?
I would have guessed the answer should be “yes” for any concept that we’re certain actually maps to a set (like the set of ‘fish’). Even if we do have fuzzy boundaries for a category; this fuzziness could reflect our uncertainty about which set we’re pointing to or uncertainty about the rules that make the set. (Doubtless I’m echoing some 15 year stale argument...) Explicitly, uncertainty whether an olive is a fruit could reflect either one of the following problems:
The person isn’t certain which set called “fruit” you mean. Do you mean fruit1 defined as all plant organs that have seeds, or fruit2 defined as plant organs that are exceptionally sweet?
The person isn’t sure what the conditions are for being in the set “fruit” and supposes that olives could belong to the set with some probability. That is, the uncertainty reflects their knowledge about the set rather than the fuzziness of the set itself.
Are we discussing whether a set maps to something ‘real’ in a person’s brain? It seems natural to think of objects belonging to or not belonging to a set based on criteria—is this not what my brain does when it thinks about a set like ‘fish’? Does my brain do this sometimes? Or is the question whether my brain does this always?
Finally, is it possible to have a concept without defining a set? … Perhaps if a concept is just pattern matching? For example, let’s suppose I see the sequence 2, 4, 2, 4. I have a ‘concept’ of a pattern that it is alternating 2s and 4s. Have I necessarily defined a set (for example, the set of patterns ABABAB...) to predict the number that comes next?
In summary, I would like to understand more clearly the conclusion regarding what cognitive science has learned about the relationship between sets and concepts.
But there aren’t any rules out there in the universe that define concepts; the concepts are in our heads. You could certainly argue for picking a strict definition (i.e. figure out exactly where a “small pile of sand” becomes a “heap of sand” using T-Rex’s method), and convince everyone that it’s the best definition… but that’s just a different (though possibly better) concept, not a more accurate representation of a set rule in either sense of the word “set”.
Focusing on the boundary between member and non-member is not typically a useful practice anyways (though it can be fun, or meta-useful as when discussing cog-sci itself). As pointed out in the OP, the parts of a concept people use the most are the ones that contain the most obvious members of the set. If you’re squirreling around in the fuzzy border-lands of “is it a heap? is it not?”, then you need to go back and figure out what question you’re actually trying to ask, instead of arguing about a concept that doesn’t have much immediate usefulness.
The set of ‘heaps’ is an interesting example of a set.
Do you suppose, given our sense of pleasant dissonance with respect to trying to identify the smallest number of elements in a heap, that one of the necessary qualities of a ‘perfect’ (most typical) heap is that a person doesn’t know how many elements there are?
Or, rather … thinking about it further—being a ‘heap’ refers to how it appears to have been formed, with items randomly piling on top of one another. And at first, it is difficult to imagine ‘heaping’ of just one thing. Yet I suppose a person could heap themselves on the floor, and then given that context, it suddenly makes sense for there to be a heap of even one thing. Say, a pair of pants thrown carelessly on a chair.
So just a few moments were needed to trace the correct conditions for being a heap, and then it makes sense. The set of ‘heaps’ is defined based on the verb (items belong to the set based on the believability that a configuration may have been formed in such a way) and not on the number of elements.
Hm, that makes sense. For a similar fuzzy-quantity-border problem to what I was thinking of originally, though, consider these other nouns:
Crowd
Bunch
Drop (of liquid)
Bundle
Shard
And so on. And that’s not even getting into adjectives where the fuzzy-border property is even more pronounced (tall, short, big, small, heavy, light).
Some of these words I have ‘concepts’ for, others I don’t. If I don’t have a concept for the word, it seems to be understood just at the association stage—I can only come up with a list of contexts where I would use the word.
If a set is defined by necessary and sufficient conditions, then having an association means that you have some sufficient conditions but you don’t know the necessary conditions. That is, you can identify a region of ideas that are in the set but you don’t know where to draw any boundaries.
At this sufficient but unknown-necessary-conditions stage, you don’t really understand what the word means (you don’t have a well-defined concept) because you need to be able to delineate something that is not in a set in order to really understand anything about what a set is. I agree that not knowing the exact boundary may not be important—but it is important that there is some known interior and some known exterior. My conclusion is that any concept that is understood at any elementary level must have both sufficient and necessary conditions.
Regarding those words which are more associations than concepts, on NPR today the following sentence stuck out as relevant:
At 1:18 Daysha Eaton says:
Hearing this statement sort of cemented a hypothesis I was considering (in a draft reply to this comment this morning) that many words don’t have real concepts behind them because their main use is to signal something. For example, I would rarely use the word ‘crowd’ when thinking to myself. I would only use the word ‘crowd’ hyperbolically, to indicate to someone that they should imagine more people than they would otherwise expect. In other words, the word augments a concept of how many and the meaning depends on a particular context of myself and a listener.
I think all words have some sort of concept behind them (for example, ‘headwaters’) but since they’re not literally used to convey that concept very often, a speaker of that language can be confused about or forget the concept behind the word.
In an internal dialogue, I would only use the word ‘crowd’ if there was literally jostling of elbows; I think that is the “original” concept for me. What does ‘crowd’ mean to you?
Hm, I see where you’re coming from, and I agree with you so far as it goes when mapping between sets and concepts. But to switch back to talking about words instead of concepts, I am not sure why you (seem to be) going with a set-based approach. Thinking of words as fuzzy regions seems more useful to me when trying to analyze how words are used, and when trying to figure out how words ought to be used.
As shown in the OP, people tend to use words not as strict sets but as regions in thing-space with more and less typical members (i.e. a bluebird is a more typical member of “birds” than a penguin or an ostrich). Though some situations will demand strict borders, in general the fuzzy region approach seems more beneficial, because it allows for quick low-level inductive reasoning. If you know that a bird is typical in bird aspects A and B (say, it has feathers, and it has a large wingspan), then you can more confidently predict that it’s typical in bird aspect C (it can fly). It’s less brittle to use probabilities in situations like these than strict boolean sufficient-or-necessary values.
You’ve convinced me on the importance of fuzzy sets. I’m sure quick low-level inductive reasoning is of vital importance, but even more immediately it seems that flexibility in meaning is useful for communication. I don’t have to hunt for the exact right word, many approximate words will do, and I can describe new concepts by stretching an old one.
However, when you write:
I don’t see how sufficient-or-necessary defined sets don’t allow fuzziness. For example, saying that a bird must have feathers (a necessary condition) and that a bluebird is a bird (sufficient condition) then there is still plenty of gray area for penguins and ostriches.
I think of sufficient conditions as being associations. We learn that penguins, ostriches and doves are all birds so those are sufficient conditions. A necessary conditions puts an edge on your set—staplers aren’t birds because they don’t have feathers.
On the other hand, a stapler that looks like a bird could be thought of as a bird to oneself. If anything has enough of the characteristics of a bird you could consider it one. For example a small plastic object with no feathers whatsoever is a ‘birdie’ because it moves like a bird.
So now I’m leaning towards agreeing. We just have associations, things are like other things if the characteristics overlap, and there aren’t any necessary conditions for deciding if something is ‘like’ something else.
Yes, strongly agreed. This idea makes me want to think of adjectives as tugging a concept-defining region of thing-space in a new direction.
That works especially well in languages like Lojban, where adjectives and nouns are not distinguished from each other. For example there is “blanu” which means “x is blue” and there is “zdani’ which means “x is a house”, and you can say “blanu zdani” (blueish-house) just as easily as “zdani blanu” (houseish-blue).
That actually is a good example of a brittle requirement, in ways that are even more directly problematic than shuttlecocks and bird-shaped staplers. What about a plucked chicken? What about a duck that, due to a genetic disease, never had feathers? What about (NOTE: This example isn’t valid re: real evolutionary history) a member of an intermediary species between pterodactyl and modern birds?
Not that it particularly affects your point, but pterodactyls are not genetic precursors to birds (they split off before the clade Dinosauria,) and feathers predate the first true dinosaurs capable of flight.
Whoops, didn’t know that, thanks.
Yeah, good point. I’m entirely convinced. Even for an apparently straight-forward category like ‘bird’, there’s not a single necessary condition you can point to. Even if there are some examples of categories with necessary conditions (I don’t know), this is evidence that the necessary conditions aren’t an intrinsic part of the way we structure a concept.
Is there a concept of beauty? Is there a set containing everything that is beautiful and nothing that is not?
Yes. Next question.
I don’t think beauty is a ‘good’ concept in every day usage. Perhaps this is because it doesn’t map to a set for me!
On the other hand, in a context where the word ‘beauty’ is being used in a specific way that I could get a handle on, I can imagine knowing for that context what concept ‘beauty’ is pointing to and being able to identify the set.
Note this doesn’t have anything to do with beauty being subjective or a 2-place word. If I know what concept to tie ‘beauty’ to, I could then know what set of things are beautiful. Note also this would be a very specific set (say, ‘Beauty2035JL’) since there would probably be many similar sets that I could immediately encounter.
Then, if I’m not mistaken, the point being made here is that our idea of beauty is not a single, specific concept, but a vague feeling that could correspond to any of myriad different concepts. Therefore, we can’t give necessary and sufficient conditions for a thing to be beautiful—unless, as you mention, we determine which concept of beauty we’re referring to.
(It sounds to me like “concept” means one thing when lukeprog says it and something else when you say it.)
All that may be true—my beef is that we don’t get that from cognitive science! Plain old philosophy gets us there.
Oh, my point was that (for me) ‘beauty’ isn’t a concept but just a token that gets bandied about. I have some faith that there is a concept behind the token when other people use the word, and I have used the word in contexts where I’ve estimated that the concept is somehow ‘close enough’ to what might be meant by beauty … but simultaneously I’m aware I’m not communicating any specific concept and probably not communicating the one I intend.
Even a vague feeling could point to a concept. If you have a vague feeling about ‘beauty’ that could correspond to any of a myriad of different concepts, then that is probably a set. (The set of all things you do tie or would tie to that feeling at that moment.)
I’m liking the idea that any concept must point to an actual set, though, and I think I’ll go with that definition in my thoughts from now on. So: a ‘concept’ is something in my brain that defines a set.
… But now it is time to go back to lukeprog’s post and check what he meant by concept and start using that token with his meaning in this context..
Or it’s any of myriad different sets.
But yes, trying to standardize our terminology will probably be helpful!
I don’t think you do have anything in your brain that defines a set, except when you formulate a precise verbal or other description of an actual mathematical set. Which is not what you’re doing when you ponder a concept like beauty.
In the instant that you think about whether something is “beautiful” the concept is a transient configuration of neuron state that at one particular point in time associates something in your memory or perception with the word “beautiful”. In which case it’s not any more meaningful than any other random association that happens to occur.
How would you tell this is what is happening instead of your brain containing a well-defined set? First note, it is not going to be consistent—you will change your mind. It’s also not going to be complete—there will be things where you don’t have any particular feeling about whether they are beautiful. Also, you will be able to say whether things are beautiful even when they aren’t a discrete thing and have boundaries just as fuzzy as the concept you relate them to (e.g. a beach with surrounding landscape). And you will do this without even noticing.
In a way, I completely agree with you. I agree with you in the sense that I think you are slicing reality in exactly the right way, and any remaining disagreement is just definitional.
Agreed, we should distinguish this random-association-type thought from the brain process (however it may be done) of mentally defining a set.
What I want to do next is say that at this random-association stage of a thought process, you don’t have a concept. Ithink you haven’t seized a concept unless you’ve defined the set. The concept, if it is lurking there, hasn’t been understood and hasn’t been ‘owned’.
Yes, I think this transience is what makes it difficult to recognize and discuss concepts correctly. For example, in several places ‘concept families’ have been mentioned, as though the primary object is a fuzzy set and at instances of thought we’re picking out particular sets from this fuzzy family. I see this reversed: sets are always specific, but our thoughts transition so fluidly from one set to a nearby set that we imagine there is a single larger fuzzy set that these sets are coming from.
For example, when we think of the set of fish, we are likely to first consider something like ‘the set of animals that look just like Nemo, but with any color variation’. (This is what was referred to as a ‘most typical’ member.) Then a few seconds later we remember sharks are also fish and throw them in. Both sets are called ‘fish’ in our minds from one moment to the next, but they were different sets and our brain can distinguish them, we just didn’t bother to track the differences as our concept of ‘fish’ evolved. So a thought process will evolve a lineage of sets SetFish1-->SetFish2-->SetFish3 over the course of a few seconds.
I think the fact that our brain can easily distinguish them all (via necessary and sufficient conditions, if not actual single-word linguistic tokens) is evidence that we understand the individual sets first, and the understanding (concept) of a ‘fuzzy set family’ comes from the observation and generalization of this lineage.
Note: I mean, this is what I think now. I’d be interested in a different paradigm for how my brain understands a concept and/or a set.
http://lesswrong.com/lw/oi/mind_projection_fallacy/
I’m voting this comment down, not because I don’t think the link is relevant or useful (because it certainly is!), but because in this context linking to it without any explanation comes across as rather rude, and is less likely to be clicked on by people who would benefit from reading it.