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Nate T.

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Everything posted by Nate T.

  1. If you really want to know, formally pi is treated is the set of all equivalence classes of equivalent Cauchy sequences of numbers-- in effect pi is defined to be *all possible sequences of rational numbers* which converge to it. As such, I guess if you like you could choose a representative sequence like (3, 3.1, 3.14, 3.141, ...) and view it as a mapping from the natural numbers into the rationals. You could do the same where the nth term of the sequence is filled by the rational number you get when you approximate pi using inscribed/circumscribed regular polygons of n + 3 sides, and these two sequences would be regarded as "the same number" since the difference between them tends to zero as you look at terms further and further along (but don't ask me to prove that for you on the spot). The class of all such sequences are regarded mentally as the number pi. You can then add, multiply, etc., these sequences by adding and multiplying their representatives term by term. This is what I mean by pi being treated as a completed object. So treated in this way you can "add" pi and sqrt(2) fine. Notice I say you can "treat" pi this way, i.e., regard it formally this way. None of this sequence business invalidates your other point that we eventually need to work with a rational approximation of pi to actually get an idea of its value, which I agree with completely. You may also object to manipulating sequences as manipulating "completed infinities", which I certainly sympathize with. But in effect the value of such a formalism is to assure us that no matter how close of an approximation we want, we can always handle manipulating real numbers as though they were rational numbers once and for all. Also this conception of the real numbers lets us talk about topology, but never mind that. Sure; I have no problem with treating pi as a variable that you eventually substitute a suitable rational number into after a certain point. But we weren't really dealing with 3.14 all along-- we were dealing with *pi* all along. This is important since otherwise we'd have to do the same calculation over again if we wanted to work with 3.14159. That'd be the frozen abstraction fallacy.
  2. I think we're saying the same thing here, but IMO it would be strange to say that an irrational number is a process in mathematical formalism, since we can treat them along with the rationals as stand-alone "completed" numbers just fine, i.e., they form an ordered field, you can manipulate them in algebraic equations, and so forth. You are correct that to really say what pi is, you need some kind of construction like inscribed/circumscribed polygons about a circle, or the Leibniz series 4 - 4/3 + 4/5 - 4/7 + ... that can be made as precise as you like in principle, whether people actually ever get around to calculating the bajillionth digit of pi or not. However, when you're done with your calculation arising from a physical problem and end up getting sqrt(2)pi/7 as your answer, you round to a decimal approximation that's appropriate to the context of your problem. In this sense real numbers are concepts of method and are not to be applied literally to physical measurements.
  3. You can't, because it would take infinitely many-- for physical measurements we're stuck with the rational numbers. That's just because to specify a real number always requires some infinite process of completion (like regarding them as infinite decimals, Cauchy sequences or Dedekind cuts) that can't actually be performed with physical measurement, as whatever you're measuring with has nonzero length and hence some error associated to it, and you can't make infinitely many measurements. So real numbers are concepts of method; applying them as lengths of physical objects is improper. This, of course, doesn't make real numbers any less important in mathematics. These measurements are contextual anyway, so infinite measurements really wouldn't be of much use. The ruler that you'd like to verify has length exactly pi is at some microscopic level a ragged collection of atoms anyway. But that's what sigfigs are for. The section "Exact Measurement and Continuity" in ItOE is a really good exploration of the issues.
  4. It's impossible to tell since it requires infinitely precise measurements to tell. Since we can't measure infinitely precisely we probably shouldn't worry about classifying objects this way. Similar problem here-- smaller than any positive volume. Can't really show this one exists either-- here you can say a material object has some nontrivial extension. I think the closest result in this direction for elementary particles is the electron, and that they only have to within 10^-22 meters or so. Well a representation of an object wouldn't be unique in any meaningful sense if you allow general homeomorphisms, for one. If you just have bijections you could "represent" a three dimensional object by a line segment, for another. Point being your correspondence can't just be a bijection, you need some fairly strong degree of continuity. Such a verification would presuppose infinitely precise measurement; you'd need to show that the map sends convergence sequences to convergent sequences.
  5. We're talking about the possibility of a material object existing. In what sense other than physical existence would a material object exist? Doesn't it contradict conservation of mass (friction created large amounts of heat that didn't flow in from elsewhere)? Certainly not something an actual fluid ought to do. I know its abandoned nowadays since it has no predictive power and the kinetic theory gives a causal account of heat. In any case, even if caloric theory was discarded as being arbitrary instead of contradictory, that doesn't change my objection against a material open sphere. Before the kinetic theory was developed caloric was a perfectly reasonable assumption, since we knew a lot about how heat and fluids both behaved. I've never seen anything with an "open boundary" running around, and it's impossible to tell whether a material object even has one, let alone the properties differentiating it from objects with "closed boundary". I don't know what you mean by "physical points." But granting them for the moment, you can't just have a bijection of mathematical points and physical points, since that would run up against all kinds of paradoxes involving cardinality being ill-behaved with regard to other measures of size and shape, as I'm sure someone with your screen name is familiar. You'd need at least the continuity of your isomorphism in order to capture what you're getting at, and then you have the same infinite precision problem that you did before.
  6. This is very much analytic/synthetic flavor-- you want to tease out the implications of "existence" via pure conceptual analysis without the "messy" details of what the physical facts happen to be. You want to discover what is true in all possible words, so to speak? That which precludes the caloric theory of heat from the concept of existence is simply that caloric has not been shown to exist. The fact that such was not known at one point in time merely means that we as humans are able to imagine materials having properties which contradict each other in certain circumstances (where the contradiction is not explicit either due to ignorance or context-dropping), not that the imagined substance in question is "metaphysically possible." Does such a conception not run against the same objection I've been raising in this thread? An appeal to an object "containing all of its limit points" is a positive claim that such an object can be known to exist, which it cannot due to the impossibility of infinitely precise measurement.
  7. Then what is the distinction between a metaphysical possibility and a physical possibility? Why not just stick with the idea of possibility if you have evidence that a certain kind of entity may exist? What does it mean for matter to be continuous-- infinitely divisible? Also I still don't know what you mean by discrete vs. continuous boundary. What do you mean by atom here? Do you mean a literal point particle, or just "that which cannot be divided further"?
  8. @aleph_0: In the context of this discussion we are taking about a physically existing object, the study of the properties of which is the science of physics. If a material object exists metaphysically, it exists physically, and you want to talk about a perfect "Platonic" boundary of a physical object you must explain what you mean by it. Yes, treating heat as a fluid contradicts the facts of how heat is known to behave, and so the caloric theory was discarded; this is because we knew enough about how both heat and fluids behave. If one were to treat atoms, say, as open spheres, one would need to give a reason why one would suspect that atoms should be modeled in this way, just as scientists observed some of the properties of heat and concluded that it behaved somewhat like a fluid. But this would require you to know how open spheres behave when they interact, and I thought we agreed before that deriving physical behavior from pure mathematical models is inappropriate. So I still think that using open spheres to model physical objects is inappropriate in a way that the caloric or the ether is not. The fact that we can't see atoms doesn't matter-- it still wouldn't make sense to talk about a literal metaphysically existing boundary of an atom, in the mathematical sense. @altonhare: I agree; this is why I object to defining perceptually evident entities using abstract topological descriptions. The notion of an "open sphere" as a literally existing object is arbitrary, much as God is because of his supposed property of omniscience. Someone talking about an object in the shape of a sphere with no boundary is making the positive claim that it has such a property, which as I mentioned before would require infinitely precise measurements to check. So because the property of being "open" is arbitrary I would go further and ask why talking about objects with open boundaries is useful at all.
  9. This makes the same error, I think. In order to show that entities really fit this description one would have to show that it is "closed", which requires showing that it contains its limit points, requiring infinitely precise measurement. Moreover, it allows aleph_0 to object (correctly) that something like a table is in fact a collection of disconnected atoms and so leads to the absurd conclusion that a table is not an entity. Requiring entities to be compact and path connected would effectively involve omniscience about certain aspects of an entity. The problem behind both of these is that entities and concepts like the boundary of an entity are contextual. A table has a well defined boundary at the macroscopic level of human sense perception, but not so much at the atomic level, and that's OK provided we define our terms correctly.
  10. I don't understand the distinction-- what is an example of a metaphysically possible object which is not physically possible? This kind of thing usually leads to an analytic/synthetic type of distinction, which I don't accept. The danger of considering objects introduced by stipulating certain properties (such as a physical open sphere, or a unicorn) is that you may in fact be talking about the empty set. You must show that some objects of the kind you consider actually exist, or you aren't really taking about anything at all. Conceptual analysis is pointless if you don't know whether your concepts have referents.
  11. @aleph_0: True, this would be a simplifying assumption, but I thought I'd assume it without loss of generality. Yes, my description was trying to get at an object composed of some material whose shape was given by the description in my last post. If that isn't what you meant by "open sphere" I'd need to know what you meant by such a thing. I need you to define your terms before I can comment. What distinguishes "discrete" boundaries from "continuous" boundaries? Even from a purely mathematical standpoint, the boundary of a set is determined by the set, and doesn't obey special rules as the set changes in time-- if you want to stipulate special rules, fine, but unless you're modeling observed phenomenon you're really just making stuff up, and you aren't really saying things about physical objects. In any case, my broader point was to object to reasoning about physical objects by assuming that they have some kind of reified mathematical boundary (discrete or continuous) and deriving case conclusions as you did earlier in this thread. Moreover, I believe that speaking about "continuous objects" is vulnerable to the same kinds of objections as "exact measurement" is.
  12. By "open sphere" I take it you mean here a region in three dimensional space of the form {x : |x - a| < r} filled with some homogeneous material? To talk about such an object, you would need to verify that it has no boundary. To do so would require infinitely precise measurements to demonstrate that your object is really open since one would need more and more precise units of measurement to verify that the sphere consists of material any arbitrary distance less than r away from a, but not exactly equal. It's that "exactly" that gets you into trouble. Since you can never actually verify that it has no boundary it's not a useful thing to consider. Put differently, other poster here are correct in asserting that an open sphere cannot literally exist as a physical object in reality-- that it has a well defined mathematical description is neither here nor there, since objects like open spheres are concepts of method. Since the law of identity prohibits infinitely precise physical measurements it is inappropriate to use these open spheres rationalistically to gain knowledge about reality. You can't derive knowledge about how things touch from pure mathematics alone without reference to reality.
  13. As someone reading the Ominous Parallels right now, blurbs like this from the Executive branch aren't helping me sleep at night...
  14. Jefferson did have an opinion as to the interpretation of "general welfare." I found this quote here:
  15. @brian0918, If you have a couple bucks to drop on a supplemental text in your study of Rand's ethics, I can't recommend Tara Smith's "Viable Values" enough as a guide to Rand's metaethics. Not only does it have an entire chapter on intrinsic values, but many of the other common objections to Rand's metaethics are disposed of. A cursory perusal of the critique you linked to suggests it'll answer most of these objections, too.
  16. Um ... wow. Any chance he's the one in your sig? But yeah, I don't think my prof was actually trying to argue that we should cast things in terms of, e.g., "the probability that Newton's laws are valid"-- it's just that we wasted a whole class discussing that article based on the idea when no one couldn't even tell me what it meant; I think that's what left a big impression on me.
  17. I'm afraid not-- that was several moves ago and I don't remember most of the other details of the article. I believe the article was about Bayesianism, though, if you're looking for substantiated instances of people abusing probability in the way I mentioned.
  18. I assume, since this is an Objectivist forum, that you're interested in sources from the Objectivist corpus. Peikoff's article "The Analytic/Synthetic Dichotomy," printed in the second edition of ITOE, discusses the notion of concievability versus possibility (albeit in a slightly different context). Specifically, the section "Logic vs. Experience" has a nice discussion of why conceiving a proproposition does not constitute evidence of that proposition-- it contains his pithy summary: "Fantasy is not a form of cognition." Personally, it's a pet peeve of mine when people start misapplying the theory of probability in contexts like this. The first time I ran into it was in a philosophy of science class, where a professor gave us an article discussing at length the notion of "the probability that the theory of evolution is true." I never received an answer as to what that meant or how to calculate it as a number that wasn't ultimately picked out of the air or an implicit postulation of some sort of useless metaphysical "many-worlds" hypothesis.
  19. I mean to include myself in that, too! In any case, since the term is starting I'm going to have to bow out from significant contribution here. If I could make a suggestion, I think the sticking point here is really understanding the term "threat of force" as applying to law, since it seems to be what's causing the difficulty. I suspect the use of the term "threat" in this context may just be confusing things. In any case, I hope you resolve this dilemma; it's been fruitful for me to think about these issues, so thanks!
  20. Ugh, too much fisking Let's see: I agree with the first part of this, but not the second. If "(law) is just a threat to those who would violate its edicts", no force or even threat of force is being used until after the lawbreaker initiated force on his own, in the case of proper law. That's why distinguishing between proper and improper law is important here. That is, when a businessman receives inside information about a company and acts on it, and the SEC fines him, the government was the one who initiates force right then, since the businessman didn't actually violate anyone's rights in breaking the bad law. Their personal reasons for deciding not to murder me are irrelevant to the question of whether murder is permissible in civil society. They would only be substituting their mind for another's if they used their mind in the first place to conclude that murder was warranted, which they weren't since it never is. Sounds good-- so what's the problem here? Is it the use of the term "threat" to describe possible negative repercussions of laws? I'd be more than happy to use this new terminology of "coming into play" if you're more comfortable with that-- the idea is essentially the same. Edit: Thinking about what analogies to use to describe this situation about the relationship between laws and force in society, the analogy umpire : rules : players : : government : laws : citizens seems to be enlightening.
  21. You said the following things in your last reply: Which, in your view, is the case? Is the existence of a law (1) a blanket threat of force to all immediately on ratification, or (2) is force used only when the law is upheld? It's either/or. Also, I still don't know what you mean when you say "If a law is force." See my last comments below. I was speaking offhandedly-- literally, "a threat of retribution against rights violations" ought to read "a threat of retribution against those who violate the rights of others." Since it only directly threatens those who violate its edicts, this should answer your objection. But now it seems you've taken the position that a law does not threaten a person unless they violate its edicts, which I thought was the crux of your dilemma in the first place. That would be in contradiction to (1) above. The sun isn't a moral agent, and a mugger is, so that analogy isn't helpful. Other than that I'm not sure what you're trying to say here. I will say that if someone comes to a conclusion that they ought to murder, this isn't "correct thinking," and is a rights violation regardless of how they rationalize it. Well, for one thing, the mugger's declaration is another such explication of consequences as well, but we agree that his ultimatum is an initiation of force, so something's wrong here. Certainly: if in a society no one is breaking any laws, then the government need not use force. This emphasizes the fact that laws aren't literally force, they instead use force in certain circumstances. Either you are arguing that "laws are force" in the primary literal sense, which is a category error since laws are declarations, not actual uses of force, or that the application of law does not use force, merely "states consequences" or "makes explicit cause and effect", which is false or beside the point, since we have no examples of laws that do not carry penalties ultimately reducible to physical force.
  22. JeffS, The source of our disagreement may be something that softwareNerd suggested earlier: we're using "force" in different ways. People offhandedly describe an unjust law as force since it is in fact an initiation of force by the government, and they don't call proper laws force since the government isn't behaving like a mugger when it upholds proper law. However, technically speaking, all laws prescribe force (in the sense of ultimately prescribing force at the primary, physical level) in certain circumstances, and the bad ones merely prescribe force that violate individual rights instead of protecting them. Similarly, people generally use the term "threat" in a legal context in cases like the mugger, who is giving an expression of intending to violate your rights. But if threat means "an expression of intention to inflict injury or damage", then it's true that the government is "threatening" retribution against rights violations; however, labeling such retribution as a "threat" has no bearing on the propriety of the law. No law simply describes cause and effect like a physical law-- literally this would mean that the a law is a rule that carries itself out automatically. Just because the declaration by a government to use force is institutionalized, public and impersonal does not make it any less man-made and chosen by those who craft it and carry it out. The actual police officers making the arrest for murder are choosing to do so, and are using force. Conversely, by your reasoning one could just as easily say that the mugger's demand is merely his way of letting you know what will cause him to act in one way or another. The difference is the purpose behind using the force. If by "cause and effect" here you simply mean that laws comprise a list of actions which the government stipulates to be "initiation of force" and whose violation will result in force being taken against you, and this objective listing and carrying out process makes these laws okay, I can think of lots of improper laws that bar perfectly rightful actions in advance, for example: laws against selling marijuana. In summary, I agree with the conclusion but disagree with the method. It is true that the existence of laws is not necessarily an initiation of force. My objection is your method of arriving at that conclusion by arguing that laws "aren't" force at all, which depending on how you want to read this is either a category error or clearly false.
  23. JeffS, What difference does it make if the government "might" catch you or not-- how does this make the threat of retributive force any less of a threat? The mugger also "might" kill you in that you might disable him first, or you might run away and he might miss. The important thing is the existence of the threat of force, not when it is ultimately carried out. In the end, both agents are stipulating you a choice: behave in a certain way or you will pay a price. The difference is in the behavior they want to elicit. I don't think their ability to carry out their threats is essential here. I think you're right about the fraud example-- it's not particularly illustrative, and insofar as you've reduced fraud to physical force you're correct. I'm not so sure about the "withholding information" example, since an irrational company could simply decide to be stupid and go ahead with rolling out the product with no warning label. Any unfortunate person who uses that product was not placed in a dilemma to use it by the company, but the company nonetheless initiated force against that person. I'm more interested in what you think of the third example, that of the employer "forcing" the man to decide between a badly-paying job and starvation. If the man decides that working under those conditions would damage his life and the only alternative is starvation, is the man being placed by the employer into a "false choice" that constitutes force? I don't understand at all this notion of law being force. A law is a declaration by a government stating that if certain actions are carried out, the government will apply force; laws are descriptions about how the government uses force. Depending on whether the law is just or not, the force then used will be either retributive or initiative. I mean, can you give me an example of a law which does not ultimately threaten force if some conditions are not met?
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