Comparing Size of Infinite Subset to Parent Infinity SubsetDate: 12/16/2001 at 14:45:06 From: Roger Subject: Method for comparing size of infinite subset to parent infinite set My question relates to the method used by mathematicians to compare the size of an infinite subset (say, the set of all positive, even integers) to that of a parent infinite set (say, the set of all positive integers), from which the subset is derived. Starting with the single set of all positive integers, this method entails the splitting out into a separate subset all the even integers. Then, a one-to-one pairing off of the evens in the subset with all the original integers in the original set shows that there are the same number of positive even integers as total, even and odd, positive integers in the single, original set. My question is this: If one considers the single, original set of all positive integers the "physiological" state and one wants to compare the size of the positive even integers to the total positive integers in the context of this "physiological" state, doesn't the splitting out of the evens into a separate subset (done as part of the "experimental processing") constitute an experimental artifact, which doesn't accurately reflect the original, "physiological" state? This seems especially possible given that some of the key relations in the original, "physiological" set of positive integers are removed in the "experimentally processed" subset of even integers. For instance, in the original set, each even integer, Nx, must be accompanied by Nx-1 other integers (that is, 4 must be accompanied by 1, 2 and 3). However, in the "experimentally processed" subset of even integers, each even integer Nx is not accompanied by Nx-1 other integers. This changed relation seems to be what allows the one-to-one pairing off of evens and total integers. One might argue that avoidance of experimental artifacts is not required in mathematics because mathematics is abstract/mental, and not experimental, in nature. However, it seems to me that any investigation to discover relations about a given situation, whether done in a mathematician's mind or in a test tube, is an experiment. And, in any experiment, the need to avoid changing the situation one wants to discover relationships in should be paramount. Therefore, while the infinite subset-parent infinite set size comparison method is performed in a mathematician's mind, it is still an experiment and still substantially changes the situation it is investigating as well as the results that would be obtained relative to those that would be found in the original, unchanged situation (single set of all positive integers). If you could provide an answer to the question of why this is not an artifact, I would greatly appreciate it. Thank you. Date: 12/16/2001 at 22:57:26 From: Doctor Peterson Subject: Re: Method for comparing size of infinite subset to parent infinite set Hi, Roger. The main trouble I see is that counting, in essence, is precisely a matter of isolating the set being counted and abstracting it from its context. If I count a set of apples, I don't think about the fact that some are on the table, some are in a bowl, and some are still on the tree. The number is an abstract concept that MUST be isolated from such other facts about the set being counted. I can't think of a way to define counting that would consider relations with items outside the subset being counted, or even relations within the subset. Admittedly, it might be argued that counting is an inappropriate thing to do in some particular situation, because it rips everything out of context and loses relations; but I don't think that changes the fact that if you do count a subset, you have to ignore relations. It would be meaningless for some purposes to count drops of milk spilled on the table, because the drops may coalesce or split as I wipe them up; number is not always conserved. But that doesn't invalidate the concept of counting, only the application to this case. You might observe that all of mathematics is an "experimental artifact" as you have described, because mathematics is in essence the study of abstractions. Everything we do in math involves taking something out of context and ignoring "irrelevant" features; and if it turns out to be wrong to ignore those features, then the math is inapplicable to the situation. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 12/17/2001 at 10:24:32 From: Roger Subject: Method for comparing size of infinite subset to parent infinite set Dr. Peterson, Hi. Thank you very much for your answer to my question on infinite sets. If you don't mind, I would like to make the following counterpoints: - I agree with your point that the counting of elements in a set is the counting of individual subsets at a time, but, within the original set of all positive integers, N, each of these even integer individual subsets (or objects, in computer language), Nx, has an attached attribute that it must be accompanied by Nx-1 other individual subsets. These attached attributes, or relations, are missing in the split-out subset and, therefore, the subset doesn't seem to accurately reflect the original situation. It seems to me that you can count individual integer subsets, while also keeping track of their attached attributes or relations. - In my field, biochemistry, we can study things individually (such as a cell's nucleus) while maintaining that thing/subset in its original context (the rest of the cell) and keeping track of the relations between the subset and the context. These relations are critical to understanding the interactions of the subset and the context. - Since determining the number of even integers "relative" to the total number of total integers is entirely about measuring a relation, the argument that "counting is an inappropriate thing to do in some particular situation, because it rips everything out of context and loses relations" seems to point to the fact that one needs to keep track of relations somehow in order to get an accurate reflection of the number of evens to the total number of integers. - I think in the case of the positive integers that number is conserved. Thank you again for your feedback! Roger Granet Date: 12/17/2001 at 12:03:12 From: Doctor Peterson Subject: Re: Method for comparing size of infinite subset to parent infinite set Hi, Roger. I figured you were in science, since "physiological" is not a mathematical term. I suspect that some of your philosophical issues arise from mixing together a scientific and a mathematical perspective. Of course, science has no need to deal with infinities (at least not in experimentation), and I think that is the real issue here. Infinity doesn't behave like finite quantities, and you are trying to resolve those problems using an approach that doesn't deal with infinity as well as math has been able to. I'm not quite sure where your idea of "counting of individual subsets" comes from; in my response I was talking only about counting elements of one subset, the even numbers. Perhaps we're just getting some terminology crossed, but I want to be sure we are talking about the same thing. Are you calling the individual even integers "subsets" rather than "elements," or are you talking now about counting both the subset of even numbers and the subset of odd numbers? If you think of a number as having "an attached attribute that it must be accompanied by Nx-1 other individual subsets" [do you just mean "numbers" there?], you are no longer just talking about numbers. If you want to count numbers, you just count numbers, not their attributes. That's what I meant in saying that counting (that is, in this case, making a one-to-one correspondence with the natural numbers) ignores other attributes. In fact, as you say, not only the act of counting but the designation of a subset implicitly extracts the even numbers from their context and breaks connections they have. A set is by nature nothing but a collection of individual items ("elements"). As I said, I can't imagine anything you could call "counting" that could somehow take into account the relations among the things counted. Do you count cells? The number you get will not take their context into account; it's just a number, and ignores attributes of the cells. What you probably do is to find other measures besides the mere number of cells in a dish, that do reflect their relations. Those relations don't change the number of cells; they just tell you more than the count alone does. Perhaps what you are really trying to say is that counting is inappropriate here IF we want to maintain our common-sense understanding that the even numbers are "every other number." You want to take relations into account; so you don't really want to count even numbers at all, just as you don't just count cells. I talked about "conservation of number" just as a hint that there are indeed cases where counting is not appropriate, not to say that numbers are not conserved here; but I think you have a similar problem in mind, namely that infinite sets do not conserve _ratio_, which is what you want the count to reflect. If we want to describe even numbers in a way that fits with our understanding of them as being "half of the natural numbers," then counting is not an appropriate measure. Rather, you should look at them in context by counting the ratio of even numbers to natural numbers within some interval, and letting that interval grow larger. The limiting ratio will then be 1:2, as you expect. This measures, for example, the probability that any arbitrarily chosen natural number will be even. By looking locally (within a finite interval), we prevent the even numbers from pushing out to infinity and evading the 1:2 ratio we find locally. The problem is that you expressed this as though it invalidated the fact that there are as many even numbers as whole numbers. It doesn't; it just says that that statement is misleading if you mistakenly take it to mean that the ratio of even to natural numbers is one. It isn't, and the reason is just that infinity doesn't follow the rules of numbers. The ratio of two equal infinities is not one; it is indeterminate. When we work with infinite quantities, we have to be very careful; the probability I mentioned is not calculated by taking the ratio of cardinalities, but by taking the limit of local ratios. Ultimately, I think the point is that you can "count" the even numbers in two different ways, one (the ratio) that measures the relation between the even numbers and the natural numbers locally, and one (the cardinality) that measures the relation globally. These are different because we are dealing with infinite sets, which do not behave the way we are used to with finite sets. So the idea that even numbers constitute half of the natural numbers, and the idea that there are as many even numbers as natural numbers, are both valid; they just answer different questions. And if that's what you meant, then you are right: the confusion here is partly due to the fact that counting does not keep track of (local) relationships, as the ratio does. Does that help at all? - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 12/17/2001 at 13:58:03 From: Roger Subject: Method for comparing size of infinite subset to parent infinite set Dr. Peterson, Hi. As before, I appreciate the time you've taken in responding to my questions. I'll make this my last email since we both probably have lots of other stuff to do, too! I agree that some of my questions relate to the fact that avoidance of experimental artifacts is very important in biochemistry, which is what my background is in. My comments are: - The reason I brought up counting individual subsets (where a subset was meant as any integer that's being counted) was that you had mentioned that counting, by its very nature, considers things as isolated subsets. I think you're right that I may have misunderstood. - I think you're right that I'm more interested in ratios since they reflect the relations between integers in a single set. - I think two problems with the arguments that: -- "the ratio of even to natural numbers is one. It isn't, and the reason is just that infinity doesn't follow the rules of numbers" -- "...(the ratio) that measures the relation between the even numbers and the natural numbers locally, and one (the cardinality) that measures the relation globally. These are different because we are dealing with infinite sets, which do not behave the way we are used to with finite sets." are: - Since my question is about the basic behavior of infinite sets and the validity of the idea that there are the same number of evens as all natural numbers, I think that the argument (for why there the same number of evens as total natural numbers) that "things are differernt with infinite sets and that they behave differently" may be circular. The answer that infinities behave differently can't be an answer to "do they really behave differently in regard to ratios of evens to naturals." - Even if things were different with infinite sets than finite sets, the rules of experimentation (in a test tube or in the mind) don't change, and changing the natural situation in such a way as to change results is not a good thing. Thanks again for all your comments. I sure appreciate your taking the time to help me out! Roger Date: 12/17/2001 at 17:09:18 From: Doctor Peterson Subject: Re: Method for comparing size of infinite subset to parent infinite set Hi, Roger. The "isolated subset" I referred to was the entire subset of even numbers, which, when counted, is considered apart from its relation to the numbers around it. As for "circular reasoning": My statements near the end were not circular because the two parts come from different ways of looking at infinity. You can't divide infinity by infinity precisely because it is indeterminate, as 0/0 is. If infinity times anything is infinity, then infinity divided by infinity is anything, not necessarily 1. This does not depend on Cantor's theory of infinities and countability, but simply on arithmetic. You don't have to accept any claims about the cardinality of the even numbers in order to recognize this fact. Just study limits. Furthermore, I wasn't answering the question "Do infinities behave differently?" or even "Why do infinities behave differently?" The original question I was answering was, "Does the anomaly of finding that there are as many evens as natural numbers arise because it is invalid to ignore the context of the set of numbers we are counting?" My basic answer is, "No, counting never takes the context or relations into account. The anomaly arises because you are confusing counting with ratios, and in the context of infinity these are not related as you expect." What you take as circular reasoning was not reasoning at all, but just my final plea to recognize that you are basing your thinking on faulty assumptions. You can't just assume that infinity DOES behave like finite numbers; if you do, you yourself are guilty of circular reasoning. Since you want to look at this in terms of experimentation, perhaps you can appreciate an analogy with quantum mechanics and relativity. Physicists thought that Newtonian mechanics was all we needed, until they started looking at the physical equivalents of "infinities" and "infinitesimals" (very large and small quantities) and found that they did not behave according to our common sense, which is trained in the world of normal quantities. High speeds and tiny particles don't behave like normal things. We don't like it, and can't really explain it, but we accept that the world isn't what we thought it was. I don't think that saying "that's the way it is" is circular reasoning, even though it is not satisfying, and may in fact turn out not to be true when the next theory comes along. In math, we often have to say "that's the way it is" because we've proved it. The proof is meant to be the explanation of why it is that way; if someone rejects the proof, I guess it's hard not to look circular. Finally, your last comment suggests that I have failed to make my original point about counting. We get two different results when we look at the limit of ratios versus the cardinality of sets, simply because they answer different questions, not because one is invalid. Counting by its very nature DOES modify the situation, by ignoring distinctions and context, and such abstraction is the essence of math. It's not a "bad thing," just something to be used carefully; if you want to take into account that every other number is even, you have to ask the right question. The answer to your fundamental problem, as I see it, is to stop imagining that the cardinality of the even numbers should reflect anything about their distribution. But perhaps it will help if I point out that your idea of not "changing the natural situation in such a way as to change results" is indeed important in working with infinity, in particular with infinite series. Certain (conditionally convergent) infinite series, if manipulated in improper ways (rearranging the terms so that some move out toward infinity rather than staying within bounds) can change their sum. We do have to be careful not to assume that it is valid to rearrange terms this way. And that is (at least superficially) similar to what is being done when we count even numbers. Some mathematicians have objected to Cantor's whole concept of manipulating infinite sets; but the consensus, I believe, is that it is valid. We just have to make sure we know what it means, and not confuse cardinality with other measures. Cardinality is real; it just doesn't mean for infinite sets quite what we expect it to mean from our finite experience. The concept of one-to-one correspondence is very simple, and if you accept it as a definition of what it means to have the same size, then you are forced to accept facts like those we have been discussing. If you don't, then you can just ignore the question of how many even numbers there are, because there is no answer. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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