SpookyKitty

To see the difference between a concept and a unit most clearly, I think it is of vital importance to keep two things in mind. 1) Concepts can be applied to things. Proper nouns cannot. 2) Counterfactuals. With regard to #1, Pi is definitely like a universal in the sense that it can be defined by the predicate "a number which is the ratio of a circle's circumference to its diameter." The same ratio appears in many different circle. In this case Pi is a proper noun which refers to a singular concept, but, confusingly, also refers to a single real number. This is just an ambiguity in our language, in my opinion. The sequence of digits, 3.14159265... makes no sense apart from the concept of "a number which is the ratio of a circle's circumference to its diameter." With regard to #2, although there are many concepts with just one referent in the reality that actually is, counterfactually there could have been more. For instance, it may have been that there was only one white thing in the entire universe, but "white", even though it is in only one thing in that counterfactual universe, could have been in more things had circumstances been different. This last idea is, I think, the crux of the matter. I think it also explains why the mere possession of concepts does not translate into the possession of any knowledge. Possessing the concept of whiteness would only tell you that white things may exist, not whether or not any, in fact, do exist. It also allows you to reason about white things (i.e., consider statements of the form "If there are any white things, then they are such that...") in the form of hypotheticals, but it does not alone tell you whether the antecedents of those hypotheticals actually obtain. In short, concepts allow you to reason about any possible world, and it is precisely by this power to encompass all possibilities that prevents them from ever becoming knowledge. Just contrast the possession of the concept of whiteness with the possession of a true sentence involving whiteness such as "Snow is white." One may possess the concept of whiteness in any possible world, but one may possess the knowledge that "snow is white" only in those worlds where snow is, indeed, white. TL;DR: The possession of concepts only confers understanding, not knowledge. Understanding what a statement means alone does not tell you whether it is true or false, and is therefore not the same thing as knowledge. I think I can tell you what it would mean. Given that the axiom is false, then, there must exist distinct concepts sharing all of the same predicates. If such concepts existed, then, in any possible language, there would always exist the possibility of confusing two different concepts. For instance, let us say that we have two concepts c1 and c2 such that both are represented by "a thing which is white" and all equivalent predicates. Then, if one says that "snow is white", then that statement could potentially be consistently regarded as both true and false. This is not really a contradiction, even though it seems that way, but it does make having any sort of propositional knowledge really weird and complicated (and unnecessarily so, in my opinion). I think the above should answer that question.

@StrictlyLogical One more thing I forgot to mention. Note that, unlike the proper noun 'cogisolution', I can apply the predicate "a thing which is a solution of equation 1" to its non-referents, such as 7, and obtain true statements like "7 is not a cogisolution".

I agree that that's what it comes down to. And as I've said before, what we would lose is all abstract reasoning that concerns single things. Suppose I tell you, "I just discovered that at least one 'cogisolution' exists!" (to borrow Eiuols terminology). You then ask, "Ok, what is a 'cogisolution'"? Now, unless I have an example of a 'cogisolution' there is no way for me to answer the question, but it is possible for me to know that at least one 'cogisolution' exists, nonetheless. How you ask? Well I know that every odd-degree equation has at least one real solution. Therefore, the fifth degree equation above also has at least one real solution. I am therefore justified in saying that "At least one cogisolution exists." even though I can't give an example of such. The reason that I can do this is because I have a concept that corresponds to 'cogisolution' even though I don't have any concrete examples of such. Now, the question is, how can I possibly assign a proper noun to something I can't even give a single example of? What sense does that even make? It's clear that the referents of the concept "a thing which is a solution to equation 1" are precisely those things which satisfy the predicate, even if it turns out that there is one or none at all. But how, then, could the proper noun 'cogisolution' possibly acquire its meaning independently of the concept? And what's worse is that there is and can only ever be one 'cogisolution', so how was that ever a concept in the first place if there are no concepts corresponding to single things?

No no no, I don't want to leave anything unanswered. But if you agree with the three claims above, then I do not see how the example I already provided is inadequate.

dream_weaver.... you're a smart guy. You can do it. I'm also here to answer questions.

@Eiuol I found the quote. It's in chapter 5 of ITOE:

Just found this gem in ITOE: Which is exactly what I've been trying to get at with my definition of 'concept'.

I don't give any justification, and I don't think I can. That's why I call it an axiom. The best justification I can give is that it makes sense to me and that the opposite doesn't. Yes, I think there are many ways to represent concepts without the use of language, but language is the best way, in my opinion. Yeah, then we agree, but I would still question whether or not it makes sense as a normative claim, but that's a topic for another thread. Fair enough. I'll have to dig through ITOE to find it now. Be back in a jiffy. Wasn't it you who convinced me that proper nouns and concepts with just one referent aren't the same thing in the first place? Well thank you, and I think your responses have been top notch. It would definitely help if more people were like you and willing to read and understand first and then criticize, rather than just skipping to the conclusion, disagreeing with it, and then spewing whatever random thought comes to mind. I mean, I went through a heap of trouble trying to do half the work for everyone in figuring out what a disagreement with my argument would entail but most have ignored that in favor of baseless charges of rationalism and arguments over definitions or phrasing.

Maybe that is sloppy wording, but if you'd read the thread, you would have known that concepts can also serve as units is a claim that I agree with. I also make use of such statements in several proofs, if you'd bothered to read them. Logical dependence has nothing to do with abstractions but with statements. Please look up the definition of "logical independence", because you're speaking nonsense. You are quote-mining and taking that statement out of context and drawing implications which do not follow and that I explicitly denied in that context. Subjects can also refer to concepts. Again, please read the thread. Please explain further, this doesn't make sense, but it seems like there's a genuine point here. Sure, that can be clarified. It should say "It is possible to apply the concept represented by the predicate "a red planet" .... " but I would have guessed that the reader would already know the intended meaning without my having to use this extremely cumbersome wording. 1) Well why wouldn't it be? 2) It connects the thing represented by the subject to a statement which contains the relevant subject. Yeah, this wording should be fixed. It's not rationalism, it's logic. Sure, two statements A and B for which "A if and only if B" is logically valid cannot be philosophically neutral towards each other, but that doesn't mean that the determination itself is somehow dependent on A and B. This is a limitation of the English language. In English you can't gramatically say "Red(Mars)" like you can in a formal language, nor can you use the word "is" without possibly confusing it with the "is" of identity. That's why I say that the word "is" is not part of the predicate, so that one does not confuse the statements "Mars has the color red" (Red(Mars))and "Mars is identiical to the color red"(Mars = red) like in "Mars is red". What? This has already been addressed in the thread. 1) It most certainly does. Rand claims that axiomatic concepts are "presupposed by all knowledge" and one must therefore conclude that it is also presupposed by her theory of concept formation insofar as that theory is knowledge. 2) The use of the phrase "x has existence" should be read as "x exists", strictly speaking. I use that phraseology in order to not confuse it with logical quantification. A predicate is anything that can be said of anything else. It does not mean that what is said of is necessarily an attribute. Nor do I claim that existence is an attribute of things. Same as above. I think I have given satisfactory responses to all of your objections here.

No, I only said to imagine that everything lay in the same plane. I didn't "aribrarily throw in another axiom", that's just a result of your and Plasmatic's misreading. But even if we were talking about flatland, the whole hyperspace point shows that "aboveness" is still definable. For Pete's sakes, I have given a proof that the axioms I use are consistent. Read the goddamn paper, or don't bother. I doubt you know much about any of these things, so I really don't care much for your opinion on them. Again, see proof. Well thank you for explicitly saying this, because it seems that you do, in fact, doubt logic and math. Or that at least you simply don't understand what it means to deduce something from something else.

Yes, you got me. I confess. I am secretly a theist, and I sacrifice babies to Telebrion under the blood moon.

That's something I would agree with so I'm still not sure why it's being directed at me.

Well, yes that is a stance on the ontology of concepts, but I only added that in so that people would not get confused about statements such as "there exists a concept such that..." and think that I am implying that concepts somehow exist outside of minds. That's not a hidden premise, that's Axiom 12. While I would agree that concepts can be represented by things other than predicates, only predicates are actually needed. Well firstly, I don't think it's just a normative claim. Rand does not say that a concept merely should have at least two referents but that all concepts have at least two referents. Rand does not say that concepts with less than two referents have a peculiar relation to reality, but that they have no relation to reality at all. She says, I forget exactly where, that the use of any such invalid concept automatically falsifies any statement using it. And I strongly disagree with all of these views.

I must admit, when I first read these questions, I had to think it through for hours, because they are almost nonsensical to me. They are questions so fundamental to the way I think that it took a lot of effort to figure out what exactly is wrong with your questions. But there are three things: 1) Things do not "require" concepts. People require concepts. 2) Concepts are not used to refer to things. Although they can be used that way, they are not merely file folders for percepts and other concepts. 3) We need concepts to reason about things. In order to reason abstractly about the solutions of the given equation, i.e., make use of statements like "Consider the only real solution of the equation..." one needs the concept of "the only real solution to that equation". You can reason about things even if you don't know exactly what they are. And reasoning about them may help you eventually figure out what they are.

Maybe you should consult an authority on dragons? Maybe Tolkein or Rowling?

Concepts are represented by predicates (if by statements you really meant to say predicates), but that does not automatically mean that they "consist" of them. What concepts consist of is a question about the ontology of concepts, and I don't have an answer for that. I said, at most, not all statements are true. I never explicitly stated that "not all concepts are any good". I do agree with it, however, but only in the sense that not all concepts are particularly useful, and definitely not in the sense that some concepts are "invalid" or "unjustified" or "false". You switch from talking about statements to talking about concepts. Statements can be justified or unjustified, predicates and concepts cannot be. You still have not explained why it would be at all problematic to make use of an "unjustified" or "invalid" concept.

Concepts output statements, and statements do not represent concepts. Some concepts are about other concepts.

There is no contradiction. That concepts can themselves be subjects of predicates has already been explained earlier in the thread. Please read it before commenting again.

What relevance does this have to the thread?

No, I was referring to higher-dimensional spaces in general, but so what if I was?

That wasn't an ad hominem attack. That was an insult. The only appropriate response to refusing to engage with an argument on the basis of dismissing something one knows absolutely nothing about.

Depending on the sense of that question, no and yes. If by it you mean a concept with a single referent which cannot be merely named, then no, because given a concept represented by a predicate "a thing which has the property X and is the only thing with that property" could simply be named "The Thing Which Has The Property X And Is The Only Thing With That Property". If, however, you mean, now, in this nontrivial sense, a thing that satisfies a given property, is the only thing with that property, but which cannot be explicitly exemplified (given one's current state of knowledge), then yes. For example, the only real solution of the equation: x^5 - 3x^4 + 5x^3 - 15x^2 + 4x - 12 = 0. Because some things are defined only abstractly at first, and only later an example (possibly unique) is discovered. An interesting historical example is the discovery of the planet Neptune.

I think you misunderstood. I am not saying that the second example is an instance of the aboveness relation. We agree that it is not. What I am saying is that, after the book has been moved, my concept of aboveness does not change. Now, you could respond to this by saying that the concept of "aboveness" includes any currently existing instances as well as any that ever were or will be. That's fine, I sort of overlooked at. However, if, counterfactually, the book had never been above that table (nor will it ever be), my concept of aboveness would have been no different. Whereas you must say that in that counterfactual scenario, the concept of aboveness would somehow be different. Well that's because I don't agree with that statement. What I am saying is that, even if there were no dogs anywhere or anywhen in the universe, the concept of "dog" would still make sense. Just as the concept of "mermaid" makes sense even though there are no mermaids. I'm not sure why you need to add the qualifier "reality departing" to "hypothetical", but you still haven't answered the argument. HAHAHAHAHAHAH! This is an absolutely laughable and stunning display of ignorance and crankish math-skepticism! Go back to highschool, plasmatic, the adults are talking.

I understand everything except your overall point.

My conception of 'concept' comes from the fact that in Rand's epistemology, they are supposed to solve the problem of universals, hence they must be things which are said-of other things. This is the traditional view of concepts since at least the time of Aristotle. My notion of neutrality is that the negation of Rand's claims cannot be proven from just the definitions and axioms presented. This is about as neutral as anything can get without beign pointless. Ok, but that wasn't exactly what I meant. To make myself clearer. Suppose that I hold an invalid concept. What exactly does that mean? It sounds like invalid concepts would lead to misunderstanding. But I maintain that it is the lack of a concept that causes misunderstanding. Even though mermaids aren't real, the proposition "There are no mermaids" is perfectly intelligible and also true. On the other hand, ruling out propositions that 'contain' invalid concepts would basically annihilate logic, which is kinda why I argue that Rand's claim that a concept needs at least two referents should be rejected. EDIT: It's more helpful to think of a concept as kind of a mental program that constructs an epistemic representation of reality. Whether that representation is accurate or not, depends on things external to the program-concept, so concepts are not things which are either true or false.