No such thing as essential meaning.

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. — MindForged

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. ;)

I know that the professional mathematicians do not define sets in a way which assumes they must be finite collections. — MindForged

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.

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. — MindForged

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?

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 — MindForged

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?

Many philosophers insist on debates about the meaning of words. I once got into a debate with my old PhD supervisor about the worth of the mass of literature which debates the meaning of the word "knowledge". I argued that the literature was pointless because it had no bearing on any substantive philosophical issue and because the sensible resolution to the debate was just to say "some people define 'knowledge' in different ways'. He insisted that the debates were crucial to philosophy. As we went back and forth, it turned out that he - an experienced, intelligent, well-read and capable philosopher - thought that Philosophy just is debate about how to define words. When I frustratingly said to him "debates about the analysis of knowledge are just trivial quibbles about how to use the word 'knowledge'", he replied "that's just what Philosophy is, isn't it?" (!!!).

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:

I eventually realized it was in fact a disagreement in the meaning of the words involved (which just about sums up philosophical disagreement...) — 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

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. — MindForged

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".

Are "bow" (the weapon) and "bow" (the knot) the same word or different words? If different then what makes them different? The meaning. The sounds we speak and the symbols we write might not have an "essential meaning", but a different meaning means a different word, so words have an essential meaning; it's part of what makes them the words they are.

A word is a signifier. Reference is the signified. Together, they make a sign.

The same signifier can be used in two different signs.

Do you like that scheme?

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? — Banno

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.

A word is a signifier. Reference is the signified. Together, they make a sign.

The same signifier can be used in two different signs.

Do you like that scheme? — frank

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?

“When I use a word,” Humpty Dumpty said, in rather a scornful tone, “it means just what I choose it to mean—neither more nor less.” “The question is,” said Alice, “whether you can make words mean so many different things.” “The question is,” said Humpty Dumpty, “which is to be master—that’s all.”
LEWIS CARROLL (Charles L. Dodgson), Through the Looking-Glass, chapter 6, p. 205 (1934). First published in 1872.

For meaning, look to use.

Drop the "essential" for a bit.

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.

Words have referential associations, — Janus

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.

So, "the" refers to an act of indicating or specifying some things or class of things. — Janus

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.

"Could" is a name of the concept of possibility, "of" is a name of the concept of belonging or subsumption. — 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.

Thanks for clarifying.

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.

If words did not embody concepts how could we understand them?

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"?

If words did not embody concepts how could we understand them? — Janus

Try filling in the details of this argument. Why would you think this to be so?

I haven't defined concepts as mental objects — Janus

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.

I think one could make a persuasive argument that the concept two the english word "two" denotes is an essential meaning, since there is going to be the idea of two in any language, since there are distinct countable things in the world, and it's very useful to be able to say that.

If a lion could talk, it would say that a gazelle in the mouth is worth more than two in the grass.

In the movie (and book) Contact, SETI detects an alien signal because it's broadcasting a repeating sequence of prime numbers from 1 to 101. The idea being that math is universal. Also, the aliens have broadcasted at the frequency of hydrogen times PI, which combines basic chemistry with basic math, which is a frequency SETI actually listens to, and one Sagan came up with when writing the book. Any alien civilization capable of broadcasting a radio signal will also know chemistry, since they live in the same universe.

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".

So you think we can understand words even if we have no idea what they mean? — Janus

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.

Concepts are shared understandings. — 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?

Think of a person who is making a sequence of sounds. You suspect it's a foreign language.

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.

What sort of thing could a meaning be? Is it the dictionary definition? The intent of the speaker? The interpretation of the listener? — Banno

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?

What of metaphor, where the word means something that is not the meaning of the word? — Banno

Yes, 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. 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. — Banno

Trite!? LOL. Navigation systems are able to make use of words, so if you are right then they understand the words they use.

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? — 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.

I won't be drawn into the child's game of asking of any explanation that something is such and such: — Janus

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.