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yppop
Guest
Rhu,
If TAM blows right by Oreocle’s excellent translation of numbers into words without getting it, I’m betting your excellent syllogism isn’t going to convert him. He is after all mathematically agnostic!
Yppop
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If TAM blows right by Oreocle’s excellent translation of numbers into words without getting it, I’m betting your excellent syllogism isn’t going to convert him. He is after all mathematically agnostic!
Yppop
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Usagi
Guest
Is this method the way you would count anything else? I doubt it.
Let’s say your 1, 3, 5, 7, and 9 were actual physical objects. Perhaps they are some of those plastic, magnetic numerals that kids would play with to learn numbers when I was little. Let’s say your task is to count those. “Count” and “pair up with the natural numbers in order” are the same thing, as I tried to show with my example of counting the letters of the alphabet.
Are you going to pair them up your way, skipping numbers just because the plastic pieces themselves represent some (but not all) of the same numbers? Or are you going to pair them up my way, counting “1, 2, 3, 4, 5” to conclude you have five objects, even though the actual objects include a 7 and a 9 and are lacking a 2 and a 4?
Usagi
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Oreoracle
Guest
Set A: The set of all integers
Set B: The set of all words for the odd integers
The purpose of replacing the numbers with words was to avoid the temptation of thinking of one set as a subset of the other. I’ll give one more example that avoids the use of numbers in one of the sets entirely.
Suppose we construct a set which has as its members the terms in the following sequence: x, xyx, xyxyx, xyxyxyx, xyxyxyxyx,…
See the pattern? Okay, now this is important: The type of objects in the set should have no bearing on the size of the set. It makes no difference whether the elements be numbers, words, colors, or strings of letters as in our set. So let’s replace each element of this set with numbers in two different ways. First we will replace each term with the number of x’s in the term. This yields the set of positive integers. Next we replace each term with the number of letters in the term. This yields the positive odd numbers. Since we’re just replacing terms, this doesn’t affect the size of the sets. Thus the sets are of the same size.
Set B: The set of all words for the odd integers
The purpose of replacing the numbers with words was to avoid the temptation of thinking of one set as a subset of the other. I’ll give one more example that avoids the use of numbers in one of the sets entirely.
Suppose we construct a set which has as its members the terms in the following sequence: x, xyx, xyxyx, xyxyxyx, xyxyxyxyx,…
See the pattern? Okay, now this is important: The type of objects in the set should have no bearing on the size of the set. It makes no difference whether the elements be numbers, words, colors, or strings of letters as in our set. So let’s replace each element of this set with numbers in two different ways. First we will replace each term with the number of x’s in the term. This yields the set of positive integers. Next we replace each term with the number of letters in the term. This yields the positive odd numbers. Since we’re just replacing terms, this doesn’t affect the size of the sets. Thus the sets are of the same size.
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thinkandmull
Guest
That’s an argument? I’ve refuted this already. I leave this discussion the complete victor. (Even my old math teacher said he can’t refute my reasoning on this) Be an ostrich if you like. See you on another thread!
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Rhubarb
Guest
Proverbs 27:2
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Rhubarb
Guest
And to the rest of us: Proverbs 26:4
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thinkandmull
Guest
There really isn’t an argument in your paragraph. I didn’t arbitrarily put 2 between 1 and 3 or 6 between 5 and 7. That’s the reality of the number system. There are an equal number of 1-10 sets in comparing 1 2 3 4 5 6 7 8 9 10 and 1 3 5 7 9, but WITHIN those sets there are 2 4 6 8 10 in the sets of the first set only, therefore there are numbers in the first series than the second. Any arguments to the contrary amount to mathematical relativism
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Oreoracle
Guest
thinkandmull,
After reading your post I am not convinced that you have read mine. My post did advance an argument; namely, by replacing elements of the same set in two different ways, the two resulting sets must be the same size since the types of objects in a set do not affect its size. If you have a question about my reasoning then ask, but don’t dismiss an argument simply because you find the conclusion disagreeable.
After reading your post I am not convinced that you have read mine. My post did advance an argument; namely, by replacing elements of the same set in two different ways, the two resulting sets must be the same size since the types of objects in a set do not affect its size. If you have a question about my reasoning then ask, but don’t dismiss an argument simply because you find the conclusion disagreeable.
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thinkandmull
Guest
“x, xyx, xyxyx, xyxyxyx, xyxyxyxyx,…”
I don’t deny that there are as many x’s as y’s. Read my post on why there aren’t as many x’s as x’s plus y’s.
I don’t deny that there are as many x’s as y’s. Read my post on why there aren’t as many x’s as x’s plus y’s.
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thinkandmull
Guest
This reminds me of a book I once saw at the library that had visual mind games. Some guy in the early 20th century made a picture of a square that had parallel lines in it but no right angles. It hurt the mind to look at. I wanted to know how he did it though so I focused my eyes on it very hard, and low and behold I am very certain that it was really an optical illusion that he created, not proof that math is bunk
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Oreoracle
Guest
You misunderstood. I wasn’t comparing the x’s to the y’s.
Let’s try again. First, replace each term with the number of x’s in the term. This yields the sequence 1, 2, 3,… (the positive integers) since the first term has a single x and each subsequent term has an additional x.
Now we use a different replacement rule. This time we’ll replace each term with the number of letters in the term. This yields 1, 3, 5,… (the positive odd numbers) since we begin with a single letter and add two each time.
Note that distinct terms were replaced with distinct numbers, so the number of objects was preserved. Thus the set of integers and the set of odd numbers are of the same size.
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thinkandmull
Guest
That is still ignoring the problem I’ve pointed out: nothing prevents you from putting 1 3 and 5 along 1 2 and 3, but you are ignoring the existence of 2 and 4 then
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Oreoracle
Guest
Nowhere in my post did I say anything about putting 1, 3, and 5 along 1, 2, and 3.
Maybe it would be clearer with finite sets. Let’s say I have the set {dog, puppy, canine}. I’ll first replace dog with 1, puppy with 2, and canine with 3, so that the set is now {1, 2, 3}. Now I’ll use a different replacement rule. It could be many things, but I’ll replace each word with the number of letters in it, so that the set is now {3, 5, 6}.
Notice how it logically follows that the sets {1, 2, 3} and {3, 5, 6} are of the same size without doing any additional work. We don’t have to count them, or set them alongside each other, etc. Nothing of the sort. They must be of the same size since they are in one-to-one correspondence with the same set. In other words, they are each the same size as {dog, puppy, canine}, so they are the same size as each other.
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thinkandmull
Guest
read post 328. I’m getting the feeling that you are so use to thinking like Cantor that you can’t begin to see things how I am explaining them. Maybe people not as use to math would understand my arguments better than people who have exposed themselves to math for too long, like on questions like the square root of a negative 
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Tomdstone
Guest
Are you against the square root of a negative?Maybe people not as use to math would understand my arguments better than people who have exposed themselves to math for too long, like on questions like the square root of a negative
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Oreoracle
Guest
thinkandmull,
Okay, let’s focus on your views for now. Do you agree or do you disagree that there is a one-to-one correspondence between the integers and the odd numbers? Your past responses seem to indicate that you agree, but I need to be sure of that before proceeding.
Okay, let’s focus on your views for now. Do you agree or do you disagree that there is a one-to-one correspondence between the integers and the odd numbers? Your past responses seem to indicate that you agree, but I need to be sure of that before proceeding.
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thinkandmull
Guest
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thinkandmull
Guest
In Physics and Beyond (1971), Werner Heisenberg wrote: “‘There is a square foot on negative 1’ means nothing else than ‘There are important mathematical relations that are most simply represented by the introduction of the square root of negative one concept’. And yet these relations would exist even without it. That is precisely why this type of mathematics is so useful even in science and technology. What is decisive, for instance, in the theory of functions, is the existence of important mathematical laws governing the behavior of pairs of continuous variables. These relations are rendered more comprehensible by the introduction of the abstract concept of the square root of negative one, although that concept is not basically needed for our understanding, and it has no counterpart among the natural numbers… In short, mathematics introduces eve higher stages of abreaction that help us attain a coherent grasp of wider realms.”
This is what I said earlier about the square root of negative numbers. The idea may have is practical use in math or physics, but as G K Chesterton said, modern men too often take the smaller truths as the bigger truths in order to do away with the bigger truths, alone within which the smaller truths have their meaning. A square root is something times itself. Nothing times itself is a negative. That’s the big truth there
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thinkandmull
Guest
Putting dogs or violins in place of the integers really hasn’t added anything to this debate
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Tomdstone
Guest
Curious.
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