SpookyKitty

Indeed, this is the correct answer. But would this task be easier or more difficult if the US, for some reason, did not recognize the state of Israel?

One time, my sister asked me to hold her baby. I agreed, but then the baby puked on my shoulder. A clear initiation of force! So I threw that baby to the ground, stomped and punched it into pieces, spat on the corpse, set it on fire and danced over it while cackling maniacally and chanting "A is A! A is A! Muahahahaha!". I mean, what did that baby think was going to happen when it decided to attack me? Any lesser response would be self-sacrificing of me.

If the US has no right to attack Israel in this case, how then does the US keep itself safe from Israeli terrorists?

Please provide textual evidence of where I did this. It seems to me that this ephemeral equivocation exists only in your head, i.e., is a totally unjustified assumption about my argument that you made and does not exist in reality. Good. Then we agree. The crimes of an individual are not automatically the crimes of his government or his countrymen. How then is the US going to keep itself safe from such hypothetical Israeli terrorists if it has no right to attack Israel?

No, my question is not invalid, that's just your (incorrect) opinion. I'm not equivocating anything since I never stated and do not believe that acts of individual terrorism are the same as acts of state sponsored terrorism. Your mere stating that a question is invalid doesn't make it so. I am simply asking you whether or not individual acts of terrorism are sufficient to justify a nuclear strike.

So your answer is that, yes, the US does have the right to nuke Israel if an Israeli terrorist attacks it? Well, congratulations on taking down that ridiculous strawman, but you still haven't answered my original question.

Let me ask you this, if some crazy Israeli citizen committed a terrorist attack on US soil and ran away back to Tel Aviv, does the US have the right to nuke Israel?

An unlawful terrorist government? I'm confused. Are you talking about Israel here? This is false. You absolutely must always respect the rights of those trying to harm you. An entire body of criminal law exists to protect the rights of criminals and the accused. It carefully delineates what kinds of actions are justifiable or unjustifiable in the course of bringing a criminal to justice. This is not some courtesy granted out of pity or mercy, but a basic foundation of justice. Preempting all possible future threats is neither a necessary nor possible nor morally justifiable course of action.

The notion that Palestinians don't have rights is absurd. What grants rights to a being is the possession of the capacity for reason. Not the exercise of reason nor the ability to reason well, but merely the capacity. Hence, even radical Islamists, collectivists of all kinds, etc. still have rights. Before one can declare the actions of a being to be evil or immoral, it must first be granted that he is capable of choosing otherwise, i.e. that he has the capacity for reason. Hence, an animal cannot be held morally responsible for killing a human. And to call any human evil is to recognize his underlying (albeit underutilized) capacity for reason. Civilization is not a prerequisite for rights. Rights are a prerequisite for civilization. No one who denies the equal rights of another has any right to call himself "civilized".

I think you are missing the point.

If I am right, then, yes, even conceiving of a single entity would require some amount of abstraction. I think this might be the key to solving a whole bunch of Parminidean paradoxes. For example, it might help to explain what we mean when we say that the "same" apple is red at one time and green at another time. But that's a story for another day. This is definitely false. Higher characteristics cannot be predicated of the entity. They are therefore not characteristics of the entity at all. Another way to look at this is, if higher characteristics were the characteristics of the underlying entity, then it you would be able to say that an entity is a characteristic of itself. Which is nonsense because entities are not the characteristics of anything. This isn't an argument. Everything I think is "merely" an abstraction in my mind. Think about what the Axiom of Identity says. An entity just is the sum total of all of its characteristics. If you are not perceiving the whole entity at any particular moment, then you are not perceiving the entity at all. The only way to get out of the contradiction is to clarify what it means to perceive the whole entity at any particular moment. Perceiving the entity does not mean perceiving all of its characteristics, nor does it mean perceiving only some of its characteristics. As the argument in the OP proves, it must mean to perceive at least one characteristic of the entity AND to also perceive a higher order characteristic which is itself the means of finding all of the other first-order characteristics. Without being able to perceive the whole entity, it is impossible to establish that the parts of the entity are, in fact, the parts of the same whole. No. The "perception" that there is something there is implicit in the perception of anything, but it is not by itself a perception. Perception is always a perception of something specific, and never a vague identityless "something". That's what I am arguing for.

I'm not ready to address this question in its totality. In all likelihood, I think there is a plurality of basic sorts, but that's only a guess. But the OP can address the question with regards to the ontological status of objects (i.e., the stricter sense of "entity"). What I think it shows is that, if we apply the Axiom of Identity consistently, then objects are really just certain second-order characteristics. And so, any division of existents into basic sorts must exclude a category of objects. I'm not really sure what you mean. But if we go back to the original question of how it is that we can perceive a whole entity without perceiving all of its characteristics, then the answer is that, in any perception of an entity, we have, at minimum, one characteristic of that entity, and a kind of "key" to finding all the other characteristics. So, in some sense, this is perceiving the whole but without perceiving the whole in all of its potentially infinite detail. I don't think so. I will spell it out a little more clearly, and then you can try to find the circularity if you're still not convinced: P1. Every entity X is the sum total of its characteristics (Axiom of Identity) P2. We perceive an entity X. (by assumption) C3. We perceive all of X's characteristics. (by substitution of P1 in P2) C4. We perceive at least one of X's characteristics, but not all of them. (from P2 and 2 above) C5. We do not perceive all of X's characteristics. (by detachment from C4) C6. We perceive all of X's characteristics and we do not perceive all of X's characteristics. (C4 and C5) C6 is a contradiction. @Boydstun Also, a question for you: Do you believe that the Identity Axiom implies the Leibnizian principle of the identity of indiscernibles?

Scientific support of the above: So people with apperceptive agnosia can see characteristics, but are unable to put them together either at all or in the right way. That means that they have lost the ability to perceive the (-,X) characteristic. If characteristics were secondary, then people with apperceptive agnosia would be completely blind.

When we perceive an entity, then either we perceive all of its characteristics, or we perceive some of its characteristics, or we perceive none of its characteristics If 3 is true, then an entity exists separately from its characteristics, which violates the Identity Axiom. 1 seems implausible, because when we perceive an entity, we are never able to perceive the back of it at the same time as the front. Nor do we perceive its insides unless we open it up somehow. This leaves 2 which also reduces to absurdity. Since an entity just is the sum total of all of its characteristics, when we perceive an entity, it must be that we perceive all and not all of its characteristics at the same time, and so we have a contradiction. Therefore, since 1,2, and 3 above are jointly exhaustive, by modus tollens, we do not perceive entities at all. There seem to be only two obvious ways out of the paradox: 4. An entity is not simply the sum total of its characteristics 5. The perception of any characteristic of an entity is also a perception of the entity as a whole Number 4 above contradicts the Identity Axiom. That just leaves 5, which I will now show leads to an infinite regress. Since the perception of any characteristic of an entity is also a perception of the entity as a whole, that means that a perception of a characteristic x of an entity X must be analyzed more accurately as the perception of a characteristic (x,X) (which we read as "the characteristic x of X"). This should make sense because a characteristic is never just a characteristic, but always a characteristic of. But this then implies that the characteristics of an entity such as X themselves also have at least the characteristic (-, X) (the characteristic of being a characteristic of X). Characteristics now become a kind of entity (with their own characteristics) since we can predicate something of them. But now the problems mentioned above repeat. We would then be forced to posit the existence of still higher characteristics forever, and we have an infinite regress. The only way to break out of the regress is to retreat all the way back to the axioms. In the argument above, we go from The characteristic x of X to The characteristic x of X is a characteristic of X But this has the form "It (the characteristic) is". This means that a characteristic is something with its own existence and identity separate from the entity which has that characteristic. Thus, the ontological order of characteristics and entities is reversed. Characteristics are primary and entities are secondary. With this knowledge we can finally explain how it is that we see an entity (which is a whole) even if we only ever perceive some of its characteristics. Since characteristics are primary, the perception of a single characteristic x is a completed perception. The perception of the entity X comes from yet another completed perception: namely, the perception of the characteristic (-,X) of x, i.e. (x,X), which points out the entity of which x is a characteristic. In conclusion, the perception of an entity merely requires the perception of two characteristics as described above. And the paradox is resolved.

But the problem with this is that there might be ways of showing that "R is (not) a type-I unit of itself" other than trying to demonstrate the negation. You are not limited to only applying the definition.

In the following, I use the term "entity" to refer to both mental and non-mental entities. Concepts are integrations of facts. A concept integrates all of the facts about its units. But, a concept of a property P therefore necessarily also integrates any universal facts about its units which can be deduced by assuming that there is some thing which is P but without assuming any particular measurements of that thing (except for those which are necessary for being P). This is just measurement omission applied to deduction. So, for example, considering the concept of "triangle", if we assume that we have a triangle without specifying any particular side lengths, or angles, etc. we can deduce that it has three sides, and that, therefore, all triangles have three sides. Such facts are necessarily true. You can have a triangle that is red or blue or rough or smooth, but you cannot have a four-sided triangle. We will call facts of the aforementioned sort conceptual facts. Now, the properties presupposed by those conceptual facts we will say are necessarily integrated (n-integrated from now on) by the concept of P. Each property P can relate to an entity in two different ways. Firstly, an entity x might actually have the property P. For example, some particular car might actually be red. But also, an entity can n-integrate a property P. For example, the concept "triangularity" n-integrates the property of being three-sided. Obviously, concretes cannot n-integrate anything. Only concepts can. Unlike concretes, concepts can therefore relate to properties in both ways. Concepts not only n-integrate properties but also have properties. Now, something interesting happens when you look at things from the point of view of the units. When an entity has a property P, it is a unit of the concept of P. We will call this type of unit a type-I unit. But the concept of P n-integrates some other properties, and since those properties have their corresponding concepts, we will consider those concepts to be type-II units of P (this justified by the fact that concepts themselves are also entities). Type-II units are important because the type-I units of P are also type-I units of each type-II unit of P. And they are so necessarily. For example, the type-I units of the concept "triangle" are necessarily type-I units of the concept "polygon" (which is a type-II unit of "triangle"), because being a triangle necessarily implies being a polygon, while being a polygon doesn't even imply being a triangle, much less necessarily so. But as we said before, concepts can also have properties. So concepts themselves can also appear as both type-I and type-II units of other concepts. In particular, the concept "abstraction" is very interesting because it is both a type-I and a type-II unit of itself. In fact, every concept is a type-II unit of itself (something which is easily proved), but not every concept is both a type-I and a type-II unit of itself. Concepts such as "abstraction" which are both type-I and type-II units of themselves I will call philosophical concepts. Let us look closely at the concept "abstraction". Since "abstraction" is a type-I unit of "abstraction", it is abstract. Now, given an abstraction without specifying anything else about it, we can deduce that it cannot be touched. Therefore, the property of being untouchable is n-integrated by the concept "abstraction", and the concept of "untouchableness" is a type-II unit of "abstraction". It follows then that, necessarily, every abstraction is untouchable. Since "abstraction" is itself abstract, it follows that "abstraction" not only n-integrates "untouchableness" but it actually has the property of being untouchable. The above argument can be generalized to any philosophical concept and any property that the philosophical concept n-integrates. That is, each philosophical concept actually has every property that it n-integrates. But we can deduce an even stronger claim by applying this reasoning to the concept "abstraction". Then we get that if "abstraction" n-integrates some property Q, since every abstraction is a unit of abstraction, it follows that every abstraction actually has the property Q. I think that the above arguments give us a powerful means to deduce the properties of concepts. AN ASIDE ON RUSSELL'S PARADOX: Consider the concept R of concepts that are not type-I units of themselves. Is R a type-I unit of itself? Answering either way results in a contradiction. I've pointed out this problem before, but I think I now have a resolution of this paradox. The key is to reject the law of excluded middle as applied to abstracta. Therefore, with regard to the philosophical conceptness of R we must remain agnostic so as to avoid the paradox.

In your opinion, does the definition of a mathematical object always have to be finite in length?

What kinds of infinite sets do you think are valid?

Regardless of my personal opinion, I want to know what your answers to the above questions are as you understand the word "exists". When it comes to abstract objects in general, I would consider myself an intrinsicist. I believe that certain abstract objects exist as part of the nature of reality. However, when it comes to mathematical objects, I believe that they are inherent parts of the nature of a certain part of reality, the mind. And further, that they exist if and only if they can be, in some sense, "conceived of", by the mind. Figuring out what counts as a "legitimate" conception is tricky. I am not completely decided about effectivism vs intuitionism. I have a far superior understanding of classical (Platonist) mathemtics as opposed to effective mathematics, and a superior understanding of effective mathematics as opposed to intuitionistic mathematics. However, I find myself leaning more and more towards the intuitionistic side every day. But, coming back to philosophy now, I also do not believe that mathematical concepts are abstracted from concretes. Rather, I believe that they are necessary components of anything that can be called consciousness. In short, no math = no consciousness (at least not human consciousness).

1. Do you believe that the set of natural numbers, N = {0,1,2,3,4,...} exists? Reasons to disagree: Some natural numbers are far too large and the universe is far too small or too short-lived to allow such numbers to ever be written down. This is called ultrafinitism. All natural numbers exist, because any natural number could be written down in principle. The set as a whole does not, however, because you cannot write down the sequence of all natural numbers even in principle. This is called strict finitism. Reasons to agree: The set of natural numbers exists as long as it understood as potentially infinite. In principle, any finite sub-sequence of 0,1,2,3,... can always be extended by adding to it the next natural number in the sequence. In other words, an infinite set exists provided that we could, in principle, generate all of its members one-by-one, even though we cannot ever actually finish this process. This is called classical finitism. The set of natural numbers exists because it can be defined. This is called Platonism. 2. Do you believe that the set of all infinite sequences of natural numbers, N -> N, exists? I.e., the set containing: 1,2,3,4,5,6,.... 12,45,92,103,... 5,5,5,5,5,... and so on... ? Reasons to disagree: I am an ultrafinitist. Since none of the elements of this set could ever be written down, none of them exist. And since none of the members of this set exist, neither does the set as a whole. I am a strict finitist. Diddo. I am a classical finitist. I believe that some members of this set exist because there exist finite programs which generate those sequences. Most of the members of this set do not exist, and so, neither does the set as a whole. However, because the set of all programs is potentially infinite, there is an alternative infinite set of sequences of natural numbers that does exist. This is the set of sequences of natural numbers that are computable. This is called effectivism. Reasons to agree: I am a classical finitist, but I do not agree that the human mind is limited to only computable operations. Given any finite sequence of natural numbers such as "1,56,987,23" the human mind is absolutely free to choose any natural number to continue such a sequence. In principle, this could be any member of the set N -> N whatsoever. As long as we understand the set of all such sequences to be potentially infinite, we could assent to its existence. This is called intuitionism. I am a Platonist. Such a set exists because it is definable. In addition to your own thoughts, where do you think Ayn Rand would fall? Rand believed in the existence of potential infinities, so I think she would be a kind of classical finitist. But effectivist, or intuitionist? On the one hand, effectivist because I doubt she would assent to the existence of anything that could not have a finite representation in the form of a word. But on the other, she did believe in absolute free will, so she might have found intuitionism to be acceptable.

This is actually a very sexy calculation. I am impressed.

@human_murda I will try to make myself clearer. If, in the course of concept formation, one is allowed to perform arbitrary computations (I mean arbitrarily chosen but specific well-defined computations such as finding averages, as opposed to computations that don't make sense) on the measurements of the characteristics prior to any sort of measurement omission/restriction (as in your proposal), then this concedes the whole point in my favor. Because at that point, most concepts would be defined primarily by the algorithms applied to the measurements, and the measurement omission/restriction steps are just secondary steps that may or may not appear in the overall algorithm that defines the concept. Stated slightly differently, Rand held that the only constructs necessary to form any non-axiomatic concept are measurement omission and differentiation. If now further constructs (such as computing averages) are required, as in your example above, then Rand's theory is false.

It doesn't matter. Any triangle that fits the definition, whether it is isosceles or scalene is fine. Because three angles do not determine a unique triangle whereas three side lengths do. This has nothing to do with borderline cases. There are no borderline cases here. Every triangle either is almost equilateral or it isn't. For larger triangles, even small differences in degrees will be detectable with the naked eye.

@human_murda That's a valiant effort there, but your omission of scaling factor idea is an instance of circular reasoning. How do you go about forming the concept of "triangles whose side lengths are such that their average is 1 while the lengths are all within 10% of that average"? Forming even this concept is not simply a matter of keeping each side length between 0.9 and 1.1. As a counterexample to a Randian definition of such a concept, consider the equilateral triangle whose side lengths are all 1.09. The given measurements fall within the allowed ranges, but the average side length is not equal to 1. The point is that any attempted definition of almost equilateral triangle concepts which considers the side lengths independently is always subtly vulnerable to extreme-case counterexamples. Your claim that I was considering triangles that cannot possibly exist is a strawman. I did not mention the triangle inequality explicitly, but that is something that could reasonably be inferred from my consistent use of the word "triangle" and not "a triple of positive real numbers". EDIT: In addition, the scaling factor is not a characteristic of a triangle. In order to count as a measurement, a comparison with a standard must be possible. So which of the many possible average-side-length-1 triangles is the standard? Note that they are not all similar to each other. This means that whatever standard is chosen, it will fail to account for almost equilateral triangles that are not similar to the standard one.

Except for axiomatic concepts, ostensive or perceptual definitions are impossible for the simple reason that non-axiomatic concepts are never perceived, only concretes are. (Strictly speaking, axiomatic concepts are never perceived either, but rahter, they are implicit in every act of cognition) Secondly, regarding "child's definitions", we are not talking about a literal child and the way that a literal child's mind works. This is merely a metaphor. Here, as in Rand's works, it is referring to any context of knowledge in which all Man has to go on is raw concept formation. In these contexts, we might call them the frontiers of knowledge, the process of concept formation proceeds wordlessly and measurements are performed through comparisons where precise quantities cannot be given. This is the context in which brand new concepts are formed. An adult definition can only arise from integration and never from concept formation. And at no point in Rand's works are measurement-dependent ranges in these contexts ever mentioned or implied.