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Everything posted by ewv

  1. Limits

    To return to Nate's original questions on limits and computation of areas with integrals, there are several basic issues. One is what it means for a sequence to have a limit; another is what it means when the limit is a real number that does not exist in the form of a rational number representation; and another is the meaning of area as a limit. These issues are actually quite easy to understand, but can be tricky to pin down if you're not familiar with some details of the concepts and how they are related. This involves both technical and conceptual explanations to fully grasp. The conceptual explanation is what is usually omitted so mainly what you are missing is the role of a hierarchy of abstract concepts that conceptually organize the technical details and the relations between them. This brings us to the question, part of which is only implicit in what you asked: What happens when the limit, such as the area under a curve, is an irrational number, like pi*r^2 for a circle? What does it mean to be the "actual area" (whether an rational or irrational number) defined only as an infinite limit representing the inside of an infinitely smooth curved boundary that is incommensurable with square or rectangular units of area? What does it mean to have an infinite sequence that never gets to infinity in which the limit it never gets to is also an irrational number defined only by an infinite decimal that also doesn't exist? Once again the answer lies in the hierarchy of abstract concepts concerning various mathematically infinite limit processes. An irrational number refers to a sequence of rational numbers, for example the square root of two is a number which when squared equals 2 within some precision. It is an abstraction referring to a sequence of rational numbers with a certain property, and is at a higher level of abstraction than the rationals. (Refer to the chapter on abstractions from abstraction in IOE.) You often see irrational numbers described by such sequences, but that isn't enough: you also have to integrate these facts into an abstract concept, concretized by a symbol while omitting the differences in precisions. Such is the meaning of the abstract concept "square root of two", whose referents are the rational numbers in the sequence with a certain property: The facts that give rise to the concept are the property that when squared they are two to some degree of precision. It is a fundamental fact that not every convergent sequence of rationals has a limit that is a rational number. When you say that some sequence has a limit which is an irrational number, you are invoking a higher level abstraction for numbers than the concepts of integers or rationals; while in the technical and numerical meaning behind it the sequence and its limit are equivalent sequences, in effect converging to each other. But for mental economy you omit those details; you invoke the higher level concept of irrational number and say that the sequence converges to such a number. When you do this for the area under a curve by taking the sequence of partial sums of the areas of rectangles you are operating on a much higher level of abstraction than adding up finite rectangles geometrically or physically to measure an area. To make the rectangles smaller and smaller you have to create conceptual units of area that are smaller and smaller by a process of subdivision. But there are no rectangles with infinitely small sides any more than there are numbers with infinitely many decimal places of precision or smallness. The concept of area under a smooth curve is a higher level abstraction just as is an irrational number and any limit process. It is neither necessary nor possible to explain the "actual" area under every smooth curve as a sum of finite square units of area that geometrically fit under the curve. (But it is still true that if you could conceptually break up the appropriate number of squares and conceptually re-arranged the pieces to fit under the curve, they would fill the area.) Furthermore, the very idea of a smooth curve is itself an abstraction because you are omitting the width of the line as non-essential. So by the time you get to the point of expressing the area by an integral you are already dealing with abstractions in many different ways, far higher in the hierarchy than measuring areas as some number of unit squares that physically fit under the curve. To sum up, understanding the cognitive status of concepts and methods like limits, integrals and areas under curves requires more than the usual collection of commonly known technical procedures. You also have to organize them and see how they lead to objective concepts at a high level of abstraction, and understand how the concepts are related. Once you have that, the mysteries go away.
  2. Healthy Eyes and Good Vision

    If your concern is avoiding the need for distance-correcting glasses, there is a form of simple laser surgy that can in many cases correct for lens focusing deficiencies. About 10 years ago it had a reputation for working but leading in some cases to problems. I don't remember the details, but have heard more recently that it has gotten better. It may be worth checking into with suitable caution. Your first priority should be to simply go to a competent eye doctor to be checked for what common or uncommon problems may be the source of your low light and distance problems. Then ask about alternatives and otherwise research them.
  3. Temperature, Velocity and Curved Space

    This is very closely related to and dependent on Ayn Rand's insights under the heading "Exact Measurement and Continuity" in the Appendix on "Measurement, Unit, and Mathematics" in IOE.
  4. Temperature, Velocity and Curved Space

    The abstract idea of the continuum entails an "infinite" subdivision, with the actual limit on "how small" left unspecified. In any particular case --- with measurements of finite precision -- it is not meaningful to specify and distinguish relative positions (differences) closer than the degree of precision to which any one of them can be specified. The same consideration applies for a theoretical understanding. Any theory can also be meaningful only to some finite degree of precision, both because it pertains to measurements and because it has some range of objective validity -- physical theories are objective, not intrinsic. Even when we observe the continuous motion of a large object at the simple perceptual level, there is only a finite degree of precision to which we can claim it is "smooth" as we watch it go by (the lenses in our eyes have a finite resolving power, our nervous system works in finite impulses, etc.). The object does not move from an infinitesimal point to an adjacent infinitesimal point: These are mathematical abstractions -- there is no infinitesimal and we don't observe it. Neither is there any such thing as "perfect" smoothness -- whether in shape (like a circular arc) or motion -- just as there is no such thing as a "perfect", i.e., an infinitely small, point. That doesn't mean that a large object moving across our field of view ceases to have a definite state beyond our realm of perception, only that we have to be careful not to extrapolate what it is and how it acts beyond the finite precision to which we can measure positions (including what we see through elementary direct observation). I don't think there is anything profound or mysterious about this. It just emphasizes the finiteness of everything we observe and the form in which we observe it, measure it or theorize about it; distinguishes between the elementary motions that we observe as smooth versus the mathematical abstractions of points and continuity; and emphasizes that both our observations and our abstract understandings are objective, not intrinsic with some kind of infinite precision.
  5. Bush's Social Security

    The advice for saving for retirement has always been to start as soon as you possibly can with at least something because of the enormous effect of compound interest.
  6. Temperature, Velocity and Curved Space

    In addition to realizing that space and time are relational concepts, not "things", the ideas of "occupying a particular location" and "motion in reality is continuous" have to be understood within the context of the principle that every measurement and every theory are valid only to some finite precision. In particular, a "location" at a point is meaningful only to some finite precision (a point is an abstraction with the "diameter" omitted; there is no infinitismal point in reality) and so is the specification of a change in position. (Likewise for measurements of time, which are based on physical changes.) "Continuous motion" is a valid abstraction, but the idea of moving "continuously" quantified to a greater precision than is applicable in any specific case is not scientifically meaningful. But one cannot deduce from these principles of finite precision in measurement an actual "quantum of distance" (or of "time") corresponding to a kind of chunk of reality as some kinds of "things". Finite precision of measurement does not mean that something is somehow actually jumping in discrete transitions corresponding to the size of the truncated precision: Measurements, and the theories associated with them, are objective, not intrinsic. Whatever ultimate consistuents there may be, which in turn may place some limit on precision of measurements of distance or time, is beyond our current knowledge and is an entirely separate issue from a proper epistemological understanding of the nature of basic measurements and higher abstractions such as continuity.
  7. Bush's Social Security

    What had you hoped for?
  8. Bush's Social Security

    Details of the Bush proposal have yet to be released as the administration is still jockeying for position and negotiating with potential allies to see what is politically feasible. There are a lot of approaches still said to be "on the table", and we should expect the inevitable political compromises to make the plan worse as time passes. The "5%" (usually still cited as 4%), seems to mean a percentage of your income, not a percentage of the social security "contributions" you pay as a tax -- which is currently about 12% of income, including the portion counted as "paid by the employer". The remaining 8% or so of "contributions" would go into the system as it is now: a tax that goes into the general fund, which is used for all government expenses including payments to those currently collecting SS benefits. There is currently no such thing as a "lock box" containing your "contributions" set aside for your future retirement, and there never has been. The Bush proposal would change that for the part that is a personal private account. So far there doesn't seem to be anything worse in principle in the Bush proposal than what we have now. One very good feature of the plan as proposed so far is that the private portion would remain your permanent property and can become part your estate. Under the current system there are no legal guarantees for recovering anything, and when a person dies all benefits are immediately cut off -- nothing can be left to a spouse, children, etc. In the not so distant past only about half of those paying into the system lived along enough to collect anything at all. Fortunately, life expectancy is greater now (but that is part of what is driving the current "pay as you go" SS system into financial crisis). The general advice concerning SS for many decades has been to not count on it at all for your retirement -- partly because SS benefits aren't enough to retire on and partially because you have no legal guarantee to collect them at all. Invest for your retirement as if SS does not exist. If Bush succeeds and you are allowed to put some of your SS "contributions" into your own private account, that could change, but your private account will always be subject to future looting through taxes for some future government-created "crisis". But that is already true of everything you own.