Objectivism Online Forum  # Nate T.

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## Everything posted by Nate T.

1. My first concern to the set of solutions approach you give above is extraneous roots. Sometimes not all of the solutions arrived at via a solution method will actually satisfy the equation for various reasons-- you would have to take this into account somehow. My second concern is that you'd have to "flatten" expression involving nested radicals (or other multivalued functions). For instance, since we certainly want sqrt(sqrt(x)) = 4thrt(x), we'd have to have formally that {{sqrt( -sqrt(x)), sqrt(sqrt(x))}, {-sqrt(-sqrt(x)), -sqrt(sqrt(x))}} = {4thrt(x), -4thrt(x)}, which are sets of different levels, and wouldn't even be equal if flattened, since some of the roots on the left side are imaginary.
2. Well, it's not as though the "restriction" approach eliminates the other roots from contention. It's just that after one extracts a square root from real equations like x^2 = a, one really has two equations to solve: x = +root(a) and x = -root(a). The only reason one does a "restriction" is to let root(a) be a single number, so that one can break up the possibilities into separate equations. That's true; pedagogically, that kind of mistake does follow from a "restriction" approach. However, if they broke up their equations into two parts as above, there would be no problem.
3. That's interesting; you got a BS in math, and then pursued a PhD in physics? Was that difficult? Did you take a fair number of physics classes in your undergrad career? Well, if you do want to do a "multi-valued function" kind of approach, you would have to treat the sets that come out of functions like the square root function as numbers in order to make sense of equations involving radicals (and any other multivalued function like the inverse trig functions). Therefore, you'd have to extend the real number line to include every possible such multiply nested collection of sets of real numbers (in order to make sense of the arcsine of the square root of one, say). Then, you would have to find a way to denote a general member of this extension if you want to solve equations with this extension; how to do this isn't obvious, at least to me. It is something of a trade-off, but I do think that it's better to restrict (albeit awkwardly) the squaring function to a bijection in order to define a square root function instead of extending the real numbers in the way outlined above. After all, in the former case one can always describe the roots in terms of the one chosen by convention (this even holds true for complex roots). To be honest, I haven't thought out a way to extend the usual algebraic operations to relations, but I'm guessing you'd encounter messes fairly quickly. Do you have a specific kind of task in mind for which the multi-valued function approach would be better?
4. Since you're using terminology out of Category Theory, I'll ask something I should have asked up front: exactly how much mathematical knowledge do you have? I think the concept you're looking for is that of a relation. This generalizes the idea of a function by allowing that one point in the domain may be "related" to more than one point in the range, and that points in the domain need not be paired to any pints in the range at all. As for "inverses" versus "inverse functions," relations always have inverse relations, so that wouldn't be an issue. Right, but how do you overcome the algebraic closure problem I give above?
5. Depending on where you look in mathematical literature, a "mapping" is either a function or a continuous function. In either case, by the definition of a function, you cannot have a function which maps one number into "several numbers." Such a "function" mapping one number into many is called a relation. You could do this, but in practice no one ever does. If you'd like to do it, I suppose that's okay, but you yourself would have to extend the arithmetical rules operating on these pairs of real numbers, since I don't know of anyone else who has done it. One problem with this is that if you wanted to extract roots of roots, you would have pairs of pairs of real numbers, and so on. Since you have to account for every possible operation in this way (i.e., you want the algebraic operation to be closed), you would have to extend the real numbers to a set which is notationally nightmarish.
6. Extracting square roots was originally invoked as a way to solve equations which look like "x^2 = a," for x. The solution set of this equation is the set of all x which satisfy this equation. Hence, if you want "the square root of a" to mean a number and not a solution set, you have to specify (through convention) which solution to the equation is meant. Mathematicians choose the positive root and express other roots in terms of that one. Given that the squaring function is a two-to-one function (except at zero), it has no inverse, and so there is no "square root" function unless one restricts the domain of the squaring function so that it is one-to-one. People usually do this by restricting the domain of the squaring function to nonnegative values (which amounts to choosing the positive solution above). So no more of this "numbers are equal to sets" business.
7. We also have things like Drug Addict + Cocaine = Happy, too. Do you think that a life of addiction to hard drugs is a happy life? Do you think it's a good life? You're promoting Hedonism here. Also, in what sense are you "justified in believing whatever you want ... as long as it doesn't affect me"? You may be legally within your rights to do whatever you want without violating another's rights, but you are not morally justified if it means you are diminishing your own life.
8. Just because different people may choose to do different things in a moral situation doesn't mean one of them was wrong, or that each person's choice is subjective. Both people can be in the same situation and decide what is in their best interests and do different things. It just means that they have different circumstances, or operate in different contexts. Hence, ethics is personal (in that valuing presupposes a valuer), but not subjective (in the sense that you can choose options which make your life better as opposed to worse).
9. Yes, but a post isn't a work of art-- you're trying to communicate with people. Communication is best done concisely, trying to convey your idea using the simplest possible language.
10. Why the fancy verbiage? You won't be taken seriously around here if you can't express yourself in plain language.
11. I had assumed that the Lebanon Revolution had more to do with the Revolutions in the Ukrane and Georgia, since they seemed to be similarly modeled.
12. Bryan, I should explain. When mathematicians write a sum from i = 1 to i = infinity, that expression is shorthand for the limit as N goes to infinity of the sum i = 1 to i = N. This is nothing new, it's what I've been saying this whole time, that the sum of a series is the limit of the sequence of partial sums. Why the distinction between an approximation of an actual sum you can never calculate and the limit of the sums? What possible use could there be in distinguishing between them? First, it isn't the smallest conceivable number, not if you want to usual operations of addition and multiplication to apply to it. Just take half of "1/infinity." There, something smaller. Second, we aren't using the same concept of infinity. I can say things like lim(x to infinity) f(x) = M, because I know it really means: for all e > 0, there exists an N > 0 such that |x| >= N implies |f(x) - M| < e. When you use it, you can't fall back on the epsilon-delta definition of the limit. You're really injecting infinity into the number line as an actual number, not as the property of a sequence. There are systems in which that is valid, but I prefer to keep my number line more tied to the perceptual level. Just make sure to mention before you say that 0.999~ < 1 that you're a non-standard analyst. It'll be great for parties. You know what-- okay. That's fine. I'll leave it to you to figure out what 0.999~ means.
13. Hal, Yuck! No Platonist, me. I should have used the term "constructed." Hmm, that's another good point. I don't quite know what to think of the Axiom of Choice. But from a pedagogical standpoint, Zorn's Lemma takes a lot of terminology. "Every nonempty partially ordered set having the property that every chain is bounded from above has a maximal member." You'd have to teach them all of those terms, and never use any of them ever again in the calculus course. Not exactly tough for a college course, but I can see the high school kids' eyes glazing over right now. They're equivalent constructions of the real line. The Dedekind cut method is awkward when it comes to multiplication, so it's generally liked less than the Cauchy Sequence construction. Although it does depend on who you ask. Well, don't get me wrong-- infinitesimals are a lot easier to understand from the get-go. And I'm certainly not attacking non-standard analysis as being logically flawed or something. I just think that, (1) pedagogically, the introduction of infinitesimals will raise more questions than it answers. ... hmm, that's actually a pretty good point. Except the motivation between the construction of the complex numbers and infinitesimal numbers are totally different. No one wanted to recognize the validity of imaginary numbers. Even Cardano, the man who first took them seriously, called them "false roots" and dismissed them as nonsensical, until he realised that he needed them in order to get real answers. Infinitesimals, on the other hand, are backward from this. People wanted a structure that could do something for them, and they just made one up for the sake of introducing a new kind of number into the number system (albeit with some pedagogical justification). Complex numbers were the result of closing the algebraic structure of the reals under root extraction. They also simply and unify many other branches of mathematics, such as the Fundamental Theorem of Algebra, which states that any polynomial of degree n has exactly n complex roots, and integral evaluation in the complex plane using analytic continuation. Such theorems are not possible without complex numbers. They have an elegant geometric strucure in the plane, and have been used to prove construction theorems in Euclidean Geometry. Considering infinitesimal roots of polynomials and solving infinitesimal equations can be done, but isn't really fruitful. Constructing this grand systems of infinitesimals to make the calculus work is fine, but then all evidence of the method used is wiped away in our final answer as irrelevant. Infinitesimals don't apply in any other branch of mathematics that I know of-- their sole purpose is to make taking limits conceptually easier. They don't have any beautiful geometric interpretation-- in fact, as far as I know, they behave just like formal variables. I guess it comes down to not constructing anything more than what you need to get the job done.
14. Bryan, There isn't one. If you have a natural number N, you can always add one to it to get a bigger natural number. So was I. That is the definition of a real number. You're right about that, but that's beside the point. You're claiming that because every partial sum 9/10 + 9/100 + ... + 9/10^N is less than one, the number 0.999~ must also be less than one, which is a non sequitur. The problem with this is that you can't 'literally' add up an infinite list of numbers. If you try, you will never be done. What you can do, however, is determine what that sum is getting very close to. That value, which is defined to be the sum of all of the numbers in the last, is the limit of the sequence of those sums. Once again, what is this "infinity", and why are you dividing by it like it were a number? Even supposing that "1/infinity" makes sense, you haven't answered what it means for a quanitity to be infinitely small, you've only given an example. Okay, let's distinguish between two things. The final answer of each of the partial sums 9/10 9/10 + 9/100 9/10 + 9/100 + 9/1000 ... and the limit, which is the value to which these final answers tend. If you mean that each of the partial sums is less than one, then that's right. But 0.999~ isn't any of these partial sums. It's their limit. You don't need to literally sit down forever to calculate 9/10 + 9/100 + ... in order to show it equal one. All you have to do (and this is the epsilon-delta definition of the limit) is show that no matter how close to 1 you specify, you can add up enough terms of the series to get even closer. That's what it means for that limit to be equal to one. I'm not sure you understand what my proof was claiming, so I'll do it in more detail. PROPOSITION. There is no greatest real number less than one. Proof. Suppose that there is a greatest real number less than one. Call it 'a'. Consider the number b = (1 + a)/2. First it is true that a < b. Second, it is true that b < 1. But a is defined to be the greatest number less than one. Since b is less than one, it follows that a is greater than it. Therefore, a is strictly less than itself, which is a contradiction. Hence no such number a exists. 0.999~ is the limit of the sequence (0.9, 0.99, 0.999, ...). I'm a senior math major.
15. Hal, Ocam's Razor cuts both ways. You can object to the definition of the limit as being overcomplicated and that of infinitesimals being much simpler, but it also postulates the existence of a huge number of entities to create the "superreal numbers," the vast majority of which you throw out of your final answer anyway when you take the "Standard Part" of your answer at the end, effectively ignoring the infinitesimals in your answer. Think of the philosophical conundrums that'll come up when students ask what these infinitesimals are, and you cannot answer them because their existence relies on Zorn's Lemma and other such high-level construction techniques. I'd rather teach my students some first order logic than tell them how to construct an abstract field extension with the Axiom of Choice. Not to mention, for those that go on in math, Analysis uses the idea behind limits to construct the real numbers, so it's helpful to have a background in limits. The seminal idea in topology is also based upon this definition. I'll stick with the epsilon-delta definition because I believe it integrates better into the hierarchy of knowledge. With the aid of limits, infinity is defined in terms of finite quantities and first order logic, which everyone understands. Postulating the existence of elements of an enlarged real line with certain preconstructed properties which are then "interpreted" as infinitesimals strikes me as formalistic, regardless of how intuitive infinitesimals might be. To me, it's the equivalent of saying: "Well, trying to construct calculus based on solely the real numbers is too much work. That infinitesimal idea almost worked, except for that small detail that there are no least positive numbers. But I want there to be a least positive number! Why not just say there's a least positive number and see what happens?" Finally, regarding the "intuitiveness" of infinitesimals, I don't regard the concept of an infinitesimal as being coherent. I regard it as a hand-wave, as a first attempt at getting a real idea of what the concept of infinity ought to be if we're going to base it our previous understanding of numbers.