Yup, the critical word is “limit”. All this stuff about quantum nature of the universe, or close enough for practicality is missing the point, and missing it enough to still be the wrong answer to the question.
Limit theory had a lot of people for many many years puzzled and arguing. Bascially because people didn’t understand what infinite is. All of these paradoxes and difficulties come from this one very deep and elusive issue. Infinite isn’t a number in the manner that we normally work with numbers. You can’t reason about it in the same way, and you can’t do a whole range of functions on it. But there are some things that you can do. And it is here that limit theory lives, and works.
For a long time many mathematicians didn’t believe in calculus. Where “believe” is meant in the same sort of way as people believe in reality. Calculus worked, was a really important tool, and was the underpinning of a large part of practical mathematics, but this elusive problem with limits and the nature of an infinite number of subdivisions of a finite world didn’t have a solid philosophical basis, and thus calculus was regarded by one school of thought as a useful tool, but not actually true. The more modern philosophical branches of mathematics deriving from the constructivists (including intuitionalists and the finitists) worry about these exact problems still.
The fundamental issue is whether infinite is a number or a process. Is it a thing (or things) that you can use and work with, or is it the process of continuing down the numbers.
It was Cantor that first worked out most of the issues. And they are hard. He spent most of his life working on them, and was a rather tortured soul whilst doing so.
Physicists mostly don’t have a true infinite, and they mostly don’t like anything that needs one. So when you map mathematics to the real world, you can get into an interesting hybrid of answers about the nature of truth and reality. One friend of mine, who is an eminent theoretical physicst, doesn’t really believe that complex numbers “exist.” Rather they are simply a way of representing phase. He probably does think that transendentals (i.e. pi) and the algebraic numbers do exist however.
Does anyone have links to the many threads where we’ve already discussed the question of whether .999… = 1, so people can just read those and not rehash all the old discussions?
The trouble with Zeno’s Paradox is it relates only to the time and distance that the can takes before it hits the ground. Each time you halve the remaining distance, you reduce the remaining amount of time under consideration as well.
The maximum time used approaches a limit, but never reaches it. Since it never reaches it mathematically, the can never hits the ground mathematically.
However, time marches on, and slightly after that limit is almost reached, it passes it and splat!
So the problem is you are artificially limiting time to just before the impact in order to create the supposed paradox.
Forgive me, I have no doubt mathematicians have spent years studying this concept and reaching this conclusion.
But that comes off as if “we’ve just decided that it does, and it works.”
From Cecil’s answer:
*The lower primate in us still resists, saying: .999~ doesn’t really represent a number, then, but a process. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.
Nonsense. The fraction 1/3 is an ordinary number, and .333~ is the same ordinary number; an infinite series of 3s simply happens to be the only way to express said number given the limitations of decimals. *
1 = 0.99999~ because 1/3 = 0.33333~ and 2/3 = 0.66666~ ?
That seems like circular, self-fulfilling logic. Each of those decimal expressions are very, very close to the number or fraction they represent. But they aren’t absolutely equal to that number or fraction.
I do have to admit that this one messed me up:
Also from C K Dexter Haven:
The simple multiplication proof is usually sufficient:
(1) x = .99999…
(2) 10x = 9.999999…
Subtracting (2)-(1):
(3) 9x = 9, hence x = 1.
“0.99999…” repeating is just notation. Notation means whatever we take it to mean. Mathematicians have, with good reason, defined this notation to mean “The smallest quantity which is above 0.9, and above 0.99, and above 0.999, and above 0.9999, and so on” (just the same as the meaning of all other decimal expansions). In the system of quantities in which this is generally interpreted, the smallest quantity greater than 0.9, 0.99, 0.999, and so on is 1. Thus, on the standard definition, 0.9999… repeating is equal to 1. This is not any kind of metaphysical claim; it is just a matter of language and notation.
Regardless of your position on that, I would like to hear your response to post #19. People often speak of the resolution to Zeno’s paradoxes as though it depends crucially on one having familiarity with the general concept of limits of sums, but I don’t think that is necessary (though it is certainly related); the fundamental insight is only that there is no contradiction between there being an infinite sequence of points within an interval and the distance between the endpoints of that interval being finite. And that there is no contradiction in this is readily seen, even by the very posers of such questions, though they forget it, merely by examining, e.g., that within the finite interval from 0 to 1 [finite because the distance between its endpoints is |1 - 0| = 1], there are (on a standard account of unceasingly divisible fractional quantities) the infinitely many points 1/2, 1/3, 1/4, etc.
Just as there is no lowest point after 0, and no greatest point below 1, and, indeed, for every fractional number, there is no immediate predecessor nor successor, so can beer fall through space without hitting a first position after its initial one, or a penultimate position before its final one, just as it is conceptually possible for there to be no first moment in time after the release of the beer and no last moment in time before the final landing of the beer. There is no contradiction in this. There is no reason to be confused by it.
I hesitate to rehash the topic yet again, but there are two concepts. One concept is related to numerical representations in base 10. The fraction 1/3 is represented in base 10 as a repeating decimal. However, the same number is represented in base 3 quite exactly as 0.1[sub]3[/sub]. In math we say these are ways to name a number, and 1/3, 0.333… and 0.1[sub]3[/sub] are three different names for the same number. So there are certain cases that are an accident of our chosen radix. Similarly, 1/7 is an even more interesting repeating decimal but represented quite neatly as 0.1[sub]7[/sub].
However, this radix-oddity approach does not resolve the repeating 9 question, because we haven’t constructed a fraction that yields a repeating 9. Instead we pursue the mathematical arguments such as that courtesy of C K Dexter Haven.
But they are not very, very close; they *are *absolutely equal to the fraction.
Here’s what you are missing: 0.3333… doesn’t mean “a really large number of 3’s, more than I could count in my lifetime.” It means “an *infinite *number of 3’s.” Infinity is not a number, it is a concept that means it never ends. (I don’t mean to sound condescending but some people really don’t grasp that.) And 0.3 followed by an infinite number of 3’s is exactly 1/3.
I could go on but there’s really nothing I could add that wasn’t already said better in any of the threads my FAQ links to.
Resolving Zeno’s paradox (in all its forms.) As described above, all you have to do is recognise that you must apply exactly the same dividing to both distance and time. The paradox (which it isn’t) is just a flaw of reasoning, where mentally you apply different processes to time and distance. There is a deep implication here - and that is that time and space require the same treatment. But that is a different conversation.
The nature of infinite. And it is the same argument again and again. Is infinite a process or a number? If you think that it is process, and a process that somehow takes a non-zero amount of time, you are sunk. Even if you consider it to be a process that takes zero time you have deep problems. (Exactly what that means is difficult to elucidate.) However you are not alone. The mainstream view of infinite is that it is a number, not a process. So 0.999999… does not have a final digit, or a notion of a tiny residual. And it is meaningless to argue about the nature of it. (The finitists however don’t really agree. But this is only the beginning of the range of mathematics they can’t manage.)
But people really can’t get their head around the idea of a real infinite - they keep thinking about it as a process - numbers that go on forever. Note how people use the term “forever” which is a notion of time. Sort of an idea that if you had forever to do it, you could run down the list of natural numbers and converge on infinity. But that is a process, and it isn’t what infinite is considered to be.
Milossarian, why don’t you read all the threads in the FAQ that CookingWithGas created related to this subject? Every objection you can come up with (yes, every one of them) and many others are discussed in those threads. The problems that you’re having with this idea aren’t new. Really, they aren’t.
I’m not sure this is fair to say. Cantor worked out issues relating to infinite cardinalities (sizes of infinite sets). But the ideas behind the modern view of calculus (with limits) were worked out before Cantor came along, by people like Cauchy and Weierstrass.
But you’re right that, in the early days of Calculus, it “worked” but wasn’t fully understood or explained. People talked about things like “infinitessimals” that were greater than zero but smaller than any real number, or fluents and fluxions and things that were in the process of becoming zero.
One thing to keep in mind is that fractions, decimals, and theoretical distances aren’t different kinds of numbers; they’re different ways of representing (the same) real numbers. Any number that can be written as a fraction can also be written as a decimal and can also be thought of as corresponding to a (theoretical) point on the number line (and/or the distance between that point and the zero point).
Ooch, no! Decimals that terminate, or have a recurring pattern that extends forever, and fractions are the same thing - they are rational numbers (numbers that are made from a ratio of two integers) but these are not the same as the real numbers. The reals are the algebraic and transendental numbers, and the rational numbers are an infentesimally small subset of these. A rational number is always algebraic, but there are infinintly more algebrreic numbers than rational. And there are even more transendantal numbers.
For instance, the square root of two is easly proven not to be a rational number. This worried the Greeks, in particular the Pythagorean school no end.
