Do Concepts and Words Have Essential Meanings?
Wasn't sure if this should have been her in Phil. of language or in metaphysics, but whatever.
Ok, so in another thread I got into what felt like an interminable back and forth with a couple of users. After what must have been a few pages of mostly repeated points, I eventually realized it was in fact a disagreement in the meaning of the words involved (which just about sums up philosophical disagreement...). So my issue was that the conflict was about if some word crucially meant some particular thing irrespective of context. Screw it, I'll stop being vague about it.
On one hand, I know that the professional mathematicians do not define sets in a way which assumes they must be finite collections. On the other hand, I was running into a wall where the insistence was that the very meaning of "collection" entails finitude.
And this strikes me as very queer, but it's not exactly new. Somewhat similarly, Quine back in the sixties or seventies insisted that negation in logic must be explosive otherwise it's not really negation, thus one side of the debate between classical and paraconsistent logicians is merely confused (I'm somewhat simplifying; Quine was rather flirtatious with radical views about logic).
So I guess my question to discuss is this: Is there some crucial, essential meaning to words or concepts (whatever) which, if you ever define differently than, you've inherently changed the subject or something? It feels really... dumb (I hate to say that. Struggling to think of another word). Because we can have a very good reason to define the word differently (Quine makes this very point), and oftentimes we can still capture a lot of the original meaning (so it's not just a semantic game by virtue of a randomly using a different meaning). Worse, something like this seems to completely undercut the objection like so:
Okay, so sets are by definition, conceptually finite collections so any attempt to define or talk about infinite collections is incoherent on pain of contradiction. But let's create a new concept and a word to refer to it: "Schmets". Schmets are just like sets, except some schmets can have infinite members provided they are defined appropriately. So now the question is, are there sets or are there schmets? Well, since sets are, by hypothesis, necessarily finite, they aren't very useful in mathematics since nearly every standard and non-standard maths uses infinity in some form or fashion (ultrafinitism doesn't look very promising). So it seems mathematicians are using Schmets and so we can just dispense with using sets in maths.
I dunno, maybe I've missed something but this move of essentializing (it's a real word, fight me) the meaning of some word doesn't seem to really move the debate along at all unless all parties involved already agree on the same meaning. There's something to be said about redefining too much, but abuse of a method doesn't necessarily mean it isn't ever valid to use it.
Ok, so in another thread I got into what felt like an interminable back and forth with a couple of users. After what must have been a few pages of mostly repeated points, I eventually realized it was in fact a disagreement in the meaning of the words involved (which just about sums up philosophical disagreement...). So my issue was that the conflict was about if some word crucially meant some particular thing irrespective of context. Screw it, I'll stop being vague about it.
On one hand, I know that the professional mathematicians do not define sets in a way which assumes they must be finite collections. On the other hand, I was running into a wall where the insistence was that the very meaning of "collection" entails finitude.
And this strikes me as very queer, but it's not exactly new. Somewhat similarly, Quine back in the sixties or seventies insisted that negation in logic must be explosive otherwise it's not really negation, thus one side of the debate between classical and paraconsistent logicians is merely confused (I'm somewhat simplifying; Quine was rather flirtatious with radical views about logic).
So I guess my question to discuss is this: Is there some crucial, essential meaning to words or concepts (whatever) which, if you ever define differently than, you've inherently changed the subject or something? It feels really... dumb (I hate to say that. Struggling to think of another word). Because we can have a very good reason to define the word differently (Quine makes this very point), and oftentimes we can still capture a lot of the original meaning (so it's not just a semantic game by virtue of a randomly using a different meaning). Worse, something like this seems to completely undercut the objection like so:
Okay, so sets are by definition, conceptually finite collections so any attempt to define or talk about infinite collections is incoherent on pain of contradiction. But let's create a new concept and a word to refer to it: "Schmets". Schmets are just like sets, except some schmets can have infinite members provided they are defined appropriately. So now the question is, are there sets or are there schmets? Well, since sets are, by hypothesis, necessarily finite, they aren't very useful in mathematics since nearly every standard and non-standard maths uses infinity in some form or fashion (ultrafinitism doesn't look very promising). So it seems mathematicians are using Schmets and so we can just dispense with using sets in maths.
I dunno, maybe I've missed something but this move of essentializing (it's a real word, fight me) the meaning of some word doesn't seem to really move the debate along at all unless all parties involved already agree on the same meaning. There's something to be said about redefining too much, but abuse of a method doesn't necessarily mean it isn't ever valid to use it.
Comments (78)
Meanings are too slippery, too inherently viewpoint-dependent, to be concretely defined. So words are just ways to limit the scope of possible understandings to the point where they can be usefully shared.
To use words properly, you need to be willing to do two things. Accept they do intend to narrow the scope for interpretation to some habitual conceptual essence. And then also show tolerance or charity for the vagueness that must always remain.
The sharing of a viewpoint or meaning doesn't have to be exact, complete, or exhaustive. Indeed, there is no other choice than to accept a fit that is going to be fuzzy at the edges, varied in its precise boundaries, creatively open in the understandings it still admits.
So I see meanings like an unruly herd of cats that you can lock up in a room. And maybe the occasional small dog or big rat gets swept up as well. If it works out well enough for some particular purpose, then that's fine.
Of course, you think technical words need to obey tighter standards. The proper understandings would be those shared by the technical community that employs them.
And that is completely reasonable. Yet the same combination of tolerant constraint has to apply. It is Quixotic to try to give words completely defined meanings. No definition could exhaust what is essentially the open ended thing of an act of interpretation. All you can do is create some habitual limit to interpretation. And that then includes the other thing of some habitual limit where it is agreed that differences in interpretation will no longer matter.
The story is rather different once you move up to an actually mathematical level of speaking. Any scientist knows the difference between trying to understand a concept in words versus actually understanding its equations.
But is one better than the other in a fundamental way?
I think here it is interesting to point to a contrast. Ordinary language is good at taking the messy physical world and restricting our focus to some conversationally limited aspect. It suppresses all the other possibilities, but does not require their elimination.
Mathematical speech on the other hand likes to start with a completely empty world and then start to construct a space of reference. So it is not limiting what already exists. It is starting with nothing and constructing whatever there is to be spoken about. It is an axiomatic approach.
So one is messy and organic. The other is clean and mechanical. I think the greatest advantage is being able to employ both well rather than take either as being the canonical case. They can complement each other, as each has its strengths and weaknesses.
The problem with the thread you mentioned is where the difference isn't recognised - and furthermore, that the difference might have to be reconciled if maths indeed aspires to talk about real physical things.
There are lots of people who reason about the world in folk terminology. And then a lot who are trained to reason in technical terminologies. But those technical terminologies inhabit their own constructed worlds, as I say. So there is yet another step to show that the constructions really can say anything complete about the real world when they come to discuss it.
The technical approach wipes the slate clean so as to build up an understanding as a set of elements. So how does it ever discover that it missed out key possibilities? Ordinary language only sweeps all the mess under a carpet. Eventually you could still stumble across it.
So you could defend a commonsense notion of infinity, or a technically constructed notion of infinity. But especially for a scientist or philosopher, the fruitful thing would be to allow the two styles of language to play off each other - accept they are in tension for good structural reason. The definiteness of the one can complement the open creativity of the other.
Having said that, using ordinary language to create shared understandings rather than defend "alternative facts" seems too much to ask of many posters. So I can understand the basic frustration you are expressing. ;)
The word "Collection" has an important role in mathematical history because it, along with the alternative "class", was proposed as a name for a group of things (NB the folk, not algebraic meaning of group is intended here) that may or may not be a set. Thus a set is a special sort of collection or class, that obeys certain properties as laid out in the Zermelo-Frankel axioms, or the axioms of some other consistent formulation of set-theory.
In that sense, all sets are collections, but not all collections are sets. So a collection certainly need not be finite.
HOWEVER......
If we approach the issue etymologically, we get a different answer. A collection refers to that which has been collected in the past. So unless we want to postulate an infinite past (which I have no problem with, but your other interlocutors do) a collection, sticking to its etymological roots, must be finite.
For that reason, I think 'class' is a better word to use either as a synonym for set, or for a concept such that all sets are classes, but not all classes are sets. For 'class', neither the etymological nor the history-of-mathematics approach implies, however faintly, that it must be finite.
PS: I don't think it works to give in and let 'set' refer only to finite classes, and make up another word for professional mathematicians to use, because school kids use infinite sets all the time, and would not want to call them schmets. I would not be at all surprised to see questions like the following in a high-school maths exam:
Q1.Tick whichever of the following are true statements about the relationship of the set of even numbers to the set of whole numbers?
A. It is a proper subset
B. It is a subset
C. It is a superset
D. It is a proper superset
Q2. How would you describe the intersection of the set of all even numbers with the set of all multiples of three, without using the words 'even', 'two' or 'three' in any form?
Q3. How many elements are in the set that is the intersection of the set of all integers greater than -2 with the set of all integers less than or equal to 2?
A. 0
B. 2
C. 4
D. 5
E. Infinitely many
All the sets considered here, except for the answer to the third one, are infinite, and high-school kids would have no problem with that.
Looks like a confusion between sets and multisets; collection being another term for multiset.
What a word means is best understood as the role it takes in whatever we are talking about. So in mathematics, a collection has a use that is distinct from our more common use.
As for words having essential meanings - no, they do not. Indeed, it is not as clear as one might think that there is such a thing as the meaning of a word. After all, can you pull the thing that is the meaning of "collection" out so that we can examine it?
I don't see why this follows.
http://mathworld.wolfram.com/Collection.html
Are you using "collection" in this specific sense? But wouldn't that mean that a collection is a (countable) set, rather than a set being a collection?
I've never understood why anyone would come to have that conception of Philosophy, or why, if you have that conception, you would still be interested in it. Philosophy, as I understand it, just isn't about words. But it looks like you are tempted by it as well:
Quoting MindForged
To answer your question, there is no such thing as the essential meaning of a word. I don't think you have missed anything. Debates about the 'real meaning' of words often look like they are relevant in some substantial way, but as far as I can tell, they never are.
PA
Words are arbitrary. We can use any string of symbols to refer to anything. Just look at all of the different languages humans use with different strings of symbols referring to the same thing ("tree" in English and "arbol" in Spanish).
Because of our limited minds, context needs to be established for us to know what some string of symbols mean. The universe does not need context. It is just the way it is and our minds try to symbolize that with language symbolizing what is in our minds. So language can refer to the things in the world via our minds. It's just that minds are inconsistent, subjective and illogical at times, so our words can be the same.
Meaning is essentially the relationship between cause and effect. What something means is what caused it. We usually associate the meaning of words with what the user intends to convey. When we can't agree on meanings of words, then we try to look at the logic of the meaning of the words that they are using. Is it consistent with the rest of what we know? If it isn't, you can safely ignore what they have said. Just look at the "gender" identity thread where many could not come up with a consistent definition of "gender" that made any sense other than "gender" is the same as "sex".
The same signifier can be used in two different signs.
Do you like that scheme?
No no, I'm saying that in the thread we could at least agree that sets are a type of collection (the everyday meaning of the words "collection"). The disagreement was whether or not "collection" necessarily implies a finite group of objects. In the bit you quoted I'm just granting that for the sake of argument, it's not about multisets.
I would say that homonyms are different words, so a word is a sign and not a signifier.
But then do we have a word (other than "signifier") that refers to the utterance/writing that can mean different things, e.g. "bow" which can mean a weapon or a knot? Perhaps also "word"? So "word" itself is a homonym that refers to either the sign or the signifier?
LEWIS CARROLL (Charles L. Dodgson), Through the Looking-Glass, chapter 6, p. 205 (1934). First published in 1872.
Do words have meanings?
What sort of thing could a meaning be? Is it the dictionary definition? The intent of the speaker? The interpretation of the listener?
What of metaphor, where the word means something that is not the meaning of the word?
And if words do not have meaning, then they certainly do not have essential meanings.
Words have referential associations, which some would count as meanings. If, in any particular case, the user of a word intends some particular referential association(s), then would that association or those associations not count as the meaning or meanings of the word in that case?
I like the Aristotelian idea, which I have seen explained on here by @Dfpolis, that to exist means to have the capacity to act, and that essence is the existent's whole range of abilities to act. Under that view the essence of a word would be its range of possible referential associations; so there could be no one essential meanings of a word, unless the context of its use were restricted such as to reduce its possible range of associations to just one.
Do they? All of them?
So all words are actually the names of things?
That just looks wrong to me. What could "the" be the name of? What could "could" name? What is the referential associate of "of"?
Words don't refer just to things, they also refer to actions and to generalities. So, "the" refers to an act of indicating or specifying some things or class of things. "Could" is a name of the concept of possibility, "of" is a name of the concept of belonging or subsumption. Remember the referential associations of words may be mutliple.
No it doesn't. It doesn't refer to anything by itself. Join it with other words and you can use it to show that there is only one - the Queen, the cat.
Quoting Janus
Concepts, so far as I can tell, are an invention used to defend such words. The theory says: "Could" must refer to something; but what that something is, is far from apparent. So we will invent a thing and call it a "concept", and say that anything we can't find a referent for must refer to a concept...
That is, concepts are a post hoc invention intended to defend a doubtful theory of meaning.
But we have been over this before.
So jumping ahead, if "of" refers to such-and-such a concept, and concepts are mental things of some sort, how is it that "of" can refer to the very same concept in my mind as it does in your mind? How can your concept of "of" be the very same as mine?
It can't, of course. So at the least "of" must refer to the innumerable distinct concepts we each have in our heads; and this convolute post-hoc theorising is supposedly simpler than just admitting the the meaning of "of" in so far as there is such a thing, is what we do with it.
That makes no sense.
Now it seems to me that there is no correct answer to the question "Can collections be infinite?"
Rather, we can decide one way or the other. If we decide that collections cannot be infinite, then there may be consequences that differ from those that would have followed, had we decided that collections can be infinite.
But it would be underhand to claim that someone who had decided that collections can be infinite was wrong.
There is correct answer, no essence of collection to which we can look for our answer.
I haven't defined concepts as mental objects so that purported difficulty of radical subjective divergence is of your own devising. If your concept of "on" were different to mine such that no translation between them could be achieved then how could I understand what you mean when you say "the brain in the vat is on the mat"?
Try filling in the details of this argument. Why would you think this to be so?
So then, what are they?
So you think we can understand words even if we have no idea what they mean?
Concepts are shared understandings.
If a lion could talk, it would say that a gazelle in the mouth is worth more than two in the grass.
As such, the assumption is that basic math and science are universal among any technological species, otherwise, they wouldn't be capable of sending or listening for radio signals.
If an alien can broadcast, then it understands "PI".
Trite. Instead, forget about meaning, especially meanings as references, and look to what we do with the words. To understand a word is just to be able to make use of it.
Quoting Janus
OK; so the meaning of a word is the concept to which it has a referential associations; and the concept is a shared understanding. So the meaning of the word is... the shared understanding it refers to?
So what is a word's shared understanding?
What would confirm for you that the sounds can be analyzed out into words?
What would confirm for you that it's just random sounds?
Same question but it's marks on a cave wall.
Meaning is the relationship between cause and effect. In the case of language use, words mean what the speaker or writer intended to convey. If it were the dictionary definition then we couldn't use metaphors. If it were the interpretation of the listener then why do speakers say, "I didn't mean it that way.", "Or that isn't what I said." when listeners misinterpret what is said. Do listeners misinterpret? If they do, then obviously meaning cannot be how it is interpreted. What exactly is the listener interpreting if not the intent behind the speakers use of words? If meaning were the listeners' interpretation then many listeners can come up with different meanings to the same string of words, and then where would we be with meaning?
Quoting BannoYes, what of them? Metaphors are simply new ways we use symbols to refer to things based on our intent. We could say the same thing in a humorous or depressing way by using metaphors. So metaphors seem to add an extra layer of context beyond what the usual string of words that are used to say the same thing - all related to intent.
Words do not have meaning independent of their use. To use words (or to use anything for that matter), you need intent. Words can be used in many different ways, which is why intent is what the real meaning is - what they were used to convey (the users intent).
Trite!? LOL. Navigation systems are able to make use of words, so if you are right then they understand the words they use.
Quoting Banno
I won't be drawn into the child's game of asking of any explanation that something is such and such: "But what is such and such?". I have no doubt that you know, just like most everyone else knows, what a shared understanding is; what else does human culture consist of?
Even in the case of disagreements, we have a shared understanding of what we disagree about, or at least, given good will, one can be reached, no?
Understanding is made possible by conceptualization. Although I think it is true that fully reflective conceptualization is possible only with language, I think animals must also be able to conceptualize in some more primal sense.
Is understanding of language itself necessarily a linguistic act? My own experience tells me that it is not. When I think of the word 'tree' I don't describe a tree to myself, I visualize a tree. I think some animals can probably visualize objects in this kind of way, and I think this counts as a kind of proto-conceptualization, indeed it is what makes language and more sophisticated forms of reflective conceptualization possible.
The animals have their own understandings of things. This is not to say that animals entertain thoughts or hold beliefs in a kind of "mental furniture" sense, and I don't think we do either. To think that would be to commit a fallacy of misplaced concreteness.
We hold beliefs and entertain thoughts as what might appear to be "mental furniture" only in the sense that we can repeatedly say the words to ourselves; but this is just a special kind of (linguistic) thinking and believing, and not by any means the whole of it.
Yeah, we don't need to get all analytic and such. Tom Tom is a machine, and therefore doesn't understand what you are saying... except when it does. The shared understanding you talk about is exactly our capacity to make use of words. But of course you can't maintain that and still insist that there is a seperate, mystical thing which is the meaning of a word.
This cannot be right. We know from many medical conditions that some people can understand words without being able to make use of them.
Broca's aphasiacs, for example, retain use of a limited vocabulary, but this does not hinder their understanding of a much broader vocabulary:
Notice how he understands the word "sixty" perfectly well, but cannot use it.
I like that.
Lacking the ability to say it hugely hampers your ability to use it; it does not hamper your ability to understand it. Hence, understanding and ability to use cannot be the same.
Next step: unpack what it means for him to understand the concept. He knows how many folk are in the organisation, and can signal that with his hands; his understanding of "60" is demonstrated in the doing.
No, he understands the word, without impediment. Yet his use of it is restricted.
Hence, your original claim, that to understand a word is to be able to use it, cannot be right.
That's our understanding of what he said; our translation of his hand signals.
But it's a neat flip, Snakes; I will however maintain that his understanding is more than some concept in his head; that it involves his being able to signal, even without using the word, that there are sixty folk in his organisation.
If he could not signal, there would be no reason to conclude that he understood "sixty".
If someone had never used "sixty", we would have no grounds to say they understood it.
But can we conclude that they do not have such understanding?
Interesting.
I think people are probably predisposed to expect or look for meaning in the same way the mind is hard-wired to pick out human faces.
Try looking at it this way Mindforged. Let's assume that a collection may be infinite and then describe what it means to be a collection, keeping in mind that a collection may be infinite. I can think of many examples of what a collection cannot be under this stipulation, ( it cannot for instance, be a bunch of things collected together in a group, because this implies finitude), but I cannot imagine what being "a collection" could actually mean with this particular criterion. Can you help me, by describing what a collection would actually be, if we allow that collections may be infinite.
Let's just collect all the odd numbers and ignore him.
An arbitrary quantity of elements referred to as a whole and which gain membership in said whole by means of sharing a common property we pick out or by being subject to the same stipulated rule.
E.g. the collection (set) of African Americans. Membership in that set is gained by the usual means and it wouldn't make a difference if there were ten of them, ten million of them or an infinite number of them. The above definition in no way can be said to require the number of members to be finite.
My capacity to use words is my understanding, to be sure, but my capacity to use words is not my actual use of words. I can have the mental capacity and be physiclly unable to use it, or I can simply refrain from using it, obviously. Also you don't seem to be paying attention; since I already said I believe there is no essential (in the sense of "separate", "mystical") meaning of a word.
Knowing how to symbolize a concept for communication is different than knowing the concept itself. What about the fact that you can't use some foreign language that you haven't learned to communicate 60? Does that mean that you don't understand 60, or does it simply mean that you don't know the symbol in the foreign language for 60?
It would seem to me that his understanding of 60 has to do with his mathematical skills, not his skills in a particular language.
That would be a never ending task, so you'd never have that collection. Perhaps you like to think that the impossible is possible Banno, but that's contradiction.
Quoting MindForged
That's fine, but if there's an infinite amount of such an element, I don't see how this qualifies as " "quantity". Don't you know that "quantity" is defined as a measurable property of something, or the number of something" Infinite is neither a measurable property nor is it a number, so you really haven't given me a definition which allows for infinity.
Really, I wish you would give more thought to what you say MindForged. How could "infinite" signify a quantity? Any such so-called "quantity" would clearly be indefinite and therefore not a quantity at all.
I'm sure you think this, but you are wrong.
Care to prove that?
I've noticed that no one has reached the end of pi yet, why do you think that you can reach the end of the odd numbers?
Are you going to make your divine declaration "I have collected all of the odd numbers", and therefore you have collected them?
Quoting apokrisis
I am a newcomer here, but I've already seen several threads or sub-threads bemoaning the lack of clearly-defined terms, ambiguity of meaning, and so on. There seems to be an automatic assumption that this is a Bad Thing. But perhaps we should be asking why we have vaguely-defined terms and ambiguity? Is there perhaps some value in this vagueness? I think there might be.
We all use essentially the same vocabulary, grammar and syntax for informal social interaction, White House propaganda, scientific reports, romantic fiction, poetry, prayer, philosophical discussion, and so on. Our language must support a variety of ways in which words are used. Our vocabularies are in the region of 20000-40000 words. If we had a one-word-per-clearly-defined-meaning language, I suspect we could need vocabularies of 100000 or more, maybe a lot more. And maybe that would be too much for the typical human mind to hold/manage?
There are a number of possible reasons why vagueness and ambiguity might be good things, or at least pragmatically-practical things.
- Informal social interaction makes creative use of ambiguity (word-play of all types).
- Poetry relies for its very existence on creative use of terms, sometimes stretching definitions to use a word in a new way, and thereby communicate a meaning that could not otherwise be practically expressed.
- Many words start with a literal definition, then accumulate metaphorical meanings. So ambiguity supports metaphor, to a degree.
- Maybe humans like, and therefore value, general terms, with necessarily vague definitions? Maybe it makes the things they (we!) want to say, easier to express? Everyday communication often does not need to be precise/exact, perhaps to the extent that too much precision of definition would make such use too difficult?
- As above, perhaps multiple meanings for words limits the amount of different words we need to memorise. There are practical limits to the size of vocabulary most of us can deal with.
I'm sure there are other similar points, but I can't think of any more at the moment. I think these are enough to make my point, though. :chin:So maybe vagueness and ambiguity have benefits, and a positive purpose? It's worth considering, I think. :chin:
Did I miss any? Did I include anything that shouldn't be there? No!
Astonishing, i know, but there you have it.
A correlation drawn between at least two things.
Words are a part of the correlation. All attribution of meaning regarding words is drawing a correlation between the words and something else.
Witt and the speech act theorists seized upon the different ways that we attribute meaning. Clearly more than just referential.
Neither did you include anything which should be there. I didn't see any odd numbers. Where's this collection you're referring to? It's easy to speak nonsense. Behold my collection of one hundred pounds of gold nuggets! Want to buy it?
Seems that way doesn't it?
The number of natural numbers is the infinite cardinal aleph-null. Ergo, by your definition it's a quantity. QED.
"How could it?" How could it not? It's not indefinite, the members of the "set of natural numbers" never increases or decreases, it is exactly what it is and has always been.
Well, I'm not sure that's quite the case under intuitionism, where infinity is only a potential, and the only natural numbers that exist are the ones which have been stated, written down or computed.
If that's so, then under one philosophy of math, natural numbers can be added over time, in a sense that they go from potential to actual. Otherwise, one might be seen as committed to the reality of infinities and numbers we haven't constructed yet.
To be very short, one can say that words are arbitrary when they are not onomatopoeic. So, there is really no "essential" connection between the word and the signified. I invoke Foucault because he presents a critical picture of the history of the analysis of language across what he calls "epistemes." This is also related to the question of essence, as considered in phenomenology.
So, what does one mean by the "essence" of a word? That is a difficult question!
Sure but I'm assuming that none of the other users here are doing constructive mathematics. In classical, standard mathematics, what I said is completely true. I'm all for discussing alternative maths and logics (I'm a logical pluralist), but I ignored it for simplicity.
However, as a matter of fact most constructivists nowadays do accept that the set of natural numbers (and any other countably infinite set) is actually infinite.
Until you demonstrate that "set of natural numbers" is not self-contradictory, such claims are nonsense. And to say that something infinite is not indefinite clearly is contradictory. So carry on with the nonsense.
Quoting Banno
I wouldn't defend such a thing. We've been through this before ...numerous times. In fact, it's you who always argues for "essential meaning" under different names. Our last disagreement, you insisted that a proposition is an essential meaning, while I argued that the meaning must be interpreted by an individual human being; interpretation is subjective.
I don't think the meanings of words is exclusively in their usage to the extent that we can ignore what the brain is doing cognitively. There's an cognitive component to meaning which makes the usage possible. I don't know about the essential part, since language is fluid and evolves. But if you tell me a word, it does cause mental activity in me.
Infinite sets are not indefinite, why do you keep saying this as if it's an obvious fact that I've conceded? Every object that's a natural number will fall into that set once I've stipulated an intensional definition of that set. You haven't once shown it to be contradictory, you just fall back on saying that anytime you're challenged to defend your position. The set of numbers equal to or greater than zero is a perfect consistent, definitely set. If you don't understand what the members of that set are, then that's because you don't understand the definition.
Right, using 'red' to denote a color is arbitrary, but the debate is over whether words have an internal meaning in the head of speakers, or the meaning is from the language use in a community, and thus external. Therefore, the beetle in the box we can't talk about, and the impossibility of private language.
To quote MCU Thor: "All words are made up."
Nope~
You're as bad as Banno with your divine proclamation. "I have collected all the natural numbers" and therefore you have collected them. Stipulating that certain numbers fall into a certain set does not make that so. This I thought was the subject being discussed here. Does a "collection" require collecting? You seem to think that a collection can be produced by stipulation. I am asking you to explain how this could be the case, but all you are doing is going around in a circle.
This appears to be your argument. Since I can stipulated the existence of a set, therefore a collection can be produced by stipulation rather than by collecting. But the original point I made, on the other thread, is that you cannot stipulate the existence of a set, because "set" is defined by "collection", and collection requires collecting. Do you see the circularity of your begging the question?
I am asking for something more than this. Can you explain what a collection which does not require collecting actually is? Let's assume, as you suggest. that it is a collection by stipulation. So I see a number of objects, and I stipulate, those objects are a collection. What makes them a real collection rather than just an imaginary collection?
Quoting MindForged
This is the issue. Your insistence that X is a perfect, consistent, definite set, does not make it so. "Set" was defined by "collection". Now we need to determine what makes something a perfect, consistent, definite collection. If it is not the act of collecting them into a group, and demonstrating that they have been collected, then what is it? Would you argue that sharing essential properties is what makes them a collection? Who would determine which properties are essential, and which are not?
Effective discourse depends on intersubjectivity: both subjects need to associate the same concept to each word. In practice, we often don't - but we can arrive at a common definitions through discussion. Of course, there are some commonly accepted definitions for many things - even then, there can be different senses of a word.
Case in point. When you say "schmets can have infinite members" do you mean "schmets can have infinitely many members" or do you mean "schmets can have infinity as a member"?
Here's a list of mathematical objects that are potential topics of discussion, each corresponding to a more general Cantorian definition of set but which have important distinctions:
Finite collections of finite objects (e.g. {1,3,5}
Infinite collections of finite objects (e.g. the set of real numbers)
fine collections of infinite objects (e.g.: {aleph-0, aleph-1})
infinite collections of infinite objects (e.g. the set of cardinal numbers)
I gave a rule that populates members of a set, I do not literally gather abstract objects and place them somewhere.Quoting Metaphysician Undercover
You're using a colloquial usage of collection, and not even the only colloquial use of that words. People have spoken of the collection of stars in the sky, only a child would think they literally meant they gathered the stars into the sky as opposed to a condition that applies to some class of objects
Quoting Metaphysician Undercover
Name some condition which applies to all of them or just create an extensional list of the objects. It's seriously simple.Quoting Metaphysician Undercover
It's not essential properties, it's just whatever properties you declare the set to be constructed based on. The "set of all red things" is, quite obviously, populated by all the objects that have the property of being red.
OK, but the question was, how does your giving a rule populate a set? Do you apprehend the issue. Suppose I decree, as you suggest, that all red things are members of a set, the set of red things. How does this declaration make certain things members of that set, while excluding other things? What if there are some things which I would say are red but someone else would say are orange? Is it the case that since I am the one who declared the set, these things are members of the set, because I think that they are red? Don't we need an official definition of "red", an essential meaning of the word? Otherwise my declaration that all red things are members of the set of red things is meaningless, and cannot populate a set because there is nothing here to indicate what it means to be red, and no one to judge which objects fulfill that condition called being red.
Quoting MindForged
You think that it is simple to name some condition which applies to all of a number of objects? This could only work if the condition which is named had an essential meaning, an official definition, allowing that all the things could be judged according to that definition. Even then, the judgement could be mistaken. So this requires two things, essential meaning and flawless judgement. A capable human being might be able to produce a flawless judgement, but where do we get the essential meaning from? Who determines exactly what it means to be red, such that one might be able to understand this essential meaning, and judge for that condition without making a mistake.
They have inherent meaning, which is assigned by us. And, for our own purposes and convenience, we change those meanings as it suits us. Language belongs to the people, and all that.... :wink: :up:
Yes, we need a degree of consistency, otherwise communication between us would be impossible. But it is also the case that word meanings are not set in stone. They are regularly and continually updated or changed, by us, the owners of language and words. Surely all this is as it should be? :chin:
A set is defined by said rule applying to the objects in question. It has nothing to do with the process of collecting things. The properties in question are possessed by those objects whether or not I accept they do or if I call it something else. Being in the set of African Americans doesn't depend on anything to do what I call them. It doesn't necessitate an essential meaning, just a conventional one which people roughly agree picks out a certain class of objects.
I've just answered this. It has nothing to do with judgements. This is akin to saying that because I haven't looked at all the even numbers I can't declare them to be part of the set of even numbers. So long as they fall under the same rule or property specification then my directly "collecting" them is entirely optional, and really irrelevant.
You're just repeating your affirmation without answering to my objection. Isn't the act of applying a rule to objects, and determining whether or not the rule applies to them, an act of collecting the ones which the rule applies to? How could a rule be applied to an object without someone applying it to that object? As much as you might insist that "a set is defined by said rule applying to the objects in question", a rule does not apply itself to an object, it must be applied. Do you not agree that someone must apply the rule, and this involves the process of interpreting it, and judging the objects, as I described in my last post? You can't just declare that all red things are members of the set of red things, and expect therefore there is a set of red things, because someone must interpret and define "red", and judge which things fulfil the conditions in order to produce that set.
Quoting MindForged
This is nonsense. What about all those people of mixed ancestry whom some would say are African American, and others would say are not. There is no "set of African Americans" unless "African Americans" has an essential meaning by which every individual might be judged. Otherwise there'd be a number of individuals who may or may not be judged to be members of the set, and the so-called set would not be a set at all due to indefiniteness.
Quoting MindForged
No, you surely did not answer this. You've gone from saying that there could be a collection without an act of collecting, to saying that a set exists as a set because you stipulate that it does, to now saying that a rule applies to an object without actually being applied to that object. Each time, you claim that something comes into existence, (a collection, a set, the applying of rules to objects), without the act which is implied by the descriptive terms.