Nate T.

It depends what the two methods are-- some methods in mathematics have limited applicability, in that they can quickly solve simple cases but give erroneous answers to more complicated ones. Coincidentally I'm teaching partial fractions to my class in the next few days, and I'd be interested in seeing any shortcuts!

Positing that there is an answer to the question "Why does existence exists?" implies things outside existence that cause existence to exist, somehow. Taking this kind of thing literally, any such existence-causing things do exist, so they are in existence and outside of existence simultaneously, a contradiction. Not surprisingly, this kind of thing tends to come from people trying to show that there is a "lower" existence (the world around us) and a "higher" existence which causes the lower existence by ineffable means (but I don't think you hold this). Also, maybe someone could shed light on this idea of things "needing" an explanation-- I don't know in what sense something could "require" an explanation.

I think this question is unanswerable. The reason I say that is because when one asks "why?", one is looking for an explanation based on other facts and principles for something. Since explanations must involve things which exist, there's no way to "step outside of existence" in an attempt to explain existence.

Thanks, Andrew-- I hope to be there.

Hi Andrew, Before winter break I attended Dr. Lewis' talk about Islamic totalitariamism and I had thought it was announced that there would be a talk this Tuesday. Do you know whether there are two talks this week, one on Tuesday and one on Wednesday, or is there just this one? Thanks.

As far as I can tell, he's basically adjoining a new element to the extended reals and saying that anything you do to it at all returns that same symbol again. I don't see how assigning it the fancy name of "nullity" is different from simply calling such expressions indeterminate, either-- after all, anything you do to an indeterminate expression will return an indeterminate expression too, since if this weren't the case you could simply undo the operations and get that you started with something determinate. I don't particularly like his "practical" reasons for introducing this symbol. L'Hopital's rule already handles indeterminate cases fine when they arise as limits, and "the pacemaker will stop if it divides by zero" argument is pretty dumb too, since you're going to have to write exceptions in your code to handle this "nullity" thing too.

Okay, that's right. It seemed like you were arguing that such a thing would be possible, because one can "conceive" of it. Just wanted to make sure that the analytic/synthetic dichotomy wasn't rearing its head here.

DMR, Would you consider a cow that is able to jump over a 1000 foot skyscraper to be possible or impossible?

That would be funny if it weren't so sad.

Well, one wouldn't use fractions to measure those. But there are plenty of things for which it would be impractical to reduce to quanta (like pizzas and rulers, among others). And rational numbers work quite well for measuring those things. Just because some objects cannot be split up into any fraction one desires does not mean that fractions have no purpose. Well, if by "object" you mean a material existent, you're right (except for the part about it being difficult to refute). That doesn't make numbers unreal or non-objective, though, since they are concepts, which are objective (if formed correctly).

Well, another way to think about it is that one could never actually carry out such a construction in reality, since it involves (countably) infinite unions and intersections. I just look at counterintuitive things like this as the price you have to pay for a general-enough theory that's able to handle countable unions and intersections.

Rational numbers count units which are regarded as equal parts of a whole unit. "7/30" means you've cut something up (whether mentally by selective focus or physically with a knife) into 30 pieces, and you've counted out 7 of them. Or, if you like, 0.4 inches means you've subdivided an inch into a new unit (dividing the inch into ten equal parts) and have measured out 4 of them Also, what issue do you take with the objectivity of numbers?

So the paradox is the fact that there exists a set which is dense in [0, 1], but at the same time doesn't have measure 1? I guess I don't understand why the compliment of the Cantor set you've constructed is so troubling while that Cantor set itself isn't. Regardless, if it's the compliment you're worried about for whatever reason, then you're right that my example doesn't do the job, since it's not open. I'm not sure what you mean by "nature must be discrete." In any case, since mathematics deals with concepts of method, the ideas in mathematics formed to give meaningful measurements of reality need not themselves correspond to reality, if that's what you're worried about (e.g., any kind of infinity in mathematics, or the imaginary unit).

This doesn't bother me very much, but that may be because I've done a lot with analysis, and so I've incorporated paradoxical sets and functions like this into my thinking. Also, a simpler example of a set containing no intervals but with positive measure is: take the unit interval [0, 1] and remove the rational numbers. The resulting set (the irrational numbers between 0 and 1) has measure 1, even though the set contains no interval (since every open interval contains a rational number).

Here's a bad spot: Not unlike "the most perfect being imaginable" or "a horse with a horn growing out if its head," just describing something with a descriptive phrase does not guarantee its existence. This is classic primacy of consciousness. Also: This is reification. Just because I have to conceptualize something in a certain way does not give existence to the devices I need to do the picturing. In other words, no one is debating that one can in a certain sense "imagine" higher dimensions in certain ways (temperature at a point, color at a point, time, etc.), but this has nothing to do with the existence of extra spatial dimensions beyond the familiar three. Again, more primacy of consciousness.

A similar question (one that also came up naturally in the thread) I had a while ago is whether it is proper to have laws requiring food companies to label any products that may have come in contact with peanuts, since these foods can cause severe reactions in some people. Am I right to conclude that those laws are improper for the same reasons you mentioned?

Hi everyone, I was recently in a discussion on LiveJournal that centered around the following situation: Suppose that a man with severe dog allergies decides to travel by plane. After the man boards, he realizes that there are other passengers with dogs and cats on the plane (this is in reality allowed by many airlines). Even after taking his emergency antihistamines, the person has an allergic reaction which requires medical treatment. The questions: Were any rights violated here? If so, who violated the rights of whom? My thoughts are: One could on one hand regard this as some sort of "fraud" on the part of the airline (they allowed pets on board and did not give the passenger notice that they would be entering an area that is hazardous to a significant portion of the population). If this were true, it would require that airlines give notification to their passengers about common allergens that may be present on their planes (pet dander, perfume, etc.) However, I was told by other people on the thread that having pets in the passenger cabin of a plane was commonplace, and so it falls to the person with the allergy to make sure they take the necessary precautions if they decide to fly. More generally, these people claimed that in any situation like this, if one has a condition that requires need, it is the responsibility of the person to check beforehand to make sure that (1) they take any necessary precautions, if possible, and (2) make arrangements with the company. Thoughts?

aleph_0: You mentioned before that you have read some Objectivist literature. Have you read Introduction to Objectivist Epistemology? If you're looking for some of Ayn Rand's thoughts about mathematics, that would be the place to look.

How much math have you studied so far, Adam? Are you learning about the math itself or about philosophical questions about mathematics?

Even if Einstein's Theory of Relativity did seem to contradict a fundamental axiom, we could dismiss the part of the theory that contradicts the axiom out of hand. Why? Because a scientific theory is the end result of a huge process of inductive generalizations intended to describe the properties of the world around us (in this example, the precession of Mercury, or the bending of light around sufficiently massive objects)-- and in order for such a process to be valid, reason (the process of organizing knowledge using non-contradiction) must also be valid. But reason cannot succeed without accepting that entities have natures, or that A is A. However, I'm pretty sure that Relativty doesn't have the kinds of implications that your friend says it does, so there's no need to try to "refute" Einstein. I'll let someone who knows more about it say more on that point.

We don't disagree. "Exactness" depends upon a standard which is appropriate for one's purposes. I was just objecting to saying that a measurement is "exact" without reference to any standard. In ordinary language, we (properly) use the term "exact" contextually, as you point out above. My usage of the term here is in the philosphical context used by Rand in "Exact Measurement and Continuity", c.f. IOE, 2nd. ed., pretty much the "ideal mathematical" definition you give above. orangeiscool's usage suggested he meant this version of exactness rather than the ordinary language version. Of course, if he didn't mean that, he can always correct me. Right-- I only meant this to apply in a discussion of measuring length (or the height of a liquid in a glass) in a continuous universe.

One of the points that Rand makes in IOE is that measurements are contextual, so there is always an implicit error term involved in any measurment. Accordingly, even if space is not quantized and one could hypothetically measure arbitrarily small lengths, it makes no sense to say that the glass is "exactly" half full. The reason is that would imply that there is zero error in a measurement, something that can't be done when measuring lengths of physical objects since one's unit of measure has some length. However, just because one cannot measure something "exactly" (i.e., acontextually, i.e., intrinsically) doesn't mean that measurement is not possible (i.e., subjective).

I see, so the referents of the verb "will exist" is all existents that ever were, are, or will be that at one point in time did not exist but at a later point did? In other words, the measurment being omitted here is not that of "does not exist now (at present) but will", but that of "does not exist at one point in time but exists at a later point in time"? Wow, this is making my head spin. Although now that I think about it, that's how it has to be. Otherwise, the referents of the verb "will exist" would change as soon as something started existing that didn't exist before. Namely, that of existing? What is the difference between "universe" and "existence," then? When I read IOE, I thought that Rand was using them as synonyms.

Okay, so to summarize, we have to be sure not to equate the referents of the concept "existence", which does refer to everything that exists, has existed or will exist, from the referents of the verb "exist" (present), whose referents are those things that exist now as opposed to in the past or in the future. We have to do this because of the open-ended nature of concepts: one's conception of "red" would be in constant flux if the referents of "red" were just the red things that have existed up until now, negating the purpose of concepts in the first place. Right? So to be precise, one would say that some entity that does not yet exist but will "does not exist, but is still in the universe?" Something about that still seems wrong, but it may be due to the technical use of the term "universe" in this context.

Is there a place in the literature I can find a reference to this (in IOE, say)? This doesn't seem right to me-- I thought that existence is that which exists now. That is, if some entity no longer exists or does not yet exist, it doesn't exist (though you could say it existed or it will exist, respectively, but that's not the same thing). If this is wrong, would you explain why?