Comparing Size of Infinite Subset to Parent Infinity Subset

Date: 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/