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Post by Arlon10 on May 3, 2020 18:27:26 GMT
Did you know that the word "hell" in several parts of the Bible owes its origin to an early method of sanitation? There was a place just outside Jerusalem where all rubbish was discarded and burned. There are various descriptions of the etymology of the word though. It means "hole" in many other contexts and there was a hole where the rubbish was discarded.
A different sort of a rubbish heap is the list of topics on this board that are burned out.
Here are some topics it might be useless to try to raise again.
A) Is 0.99999 ... (to infinity) "equal" to one?
There are two "fair" answers. In any context of real world practical measurement there is no meaningful difference. For example 1 - that expression with just 16 of the trailing 9s number of meters that is already smaller than the radius of a proton. However In a purely philosophical context there is a difference that might have some meaning later in investigations of neutrinos. And It can represent the philosophical concept of "infinitely small" or "infinitesimal" meaning something like 1/infinity, however since division by infinity is not allowed, there is the other notation.
B) Are free will and omniscience compatible?
This dilemma is easily resolved by understanding the difference between "knowledge" and "agency." Simply "knowing" George chopped down the cherry tree (if he did) in no way implies any agency in chopping down the tree other than George's.
C) The problem of evil
The most burned out of all topics, it has long been resolved by the existence of beings with free choices. It can be obvious in this "fallen" world and might not appear in higher realms as far as we know.
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Post by Eva Yojimbo on May 3, 2020 21:11:54 GMT
A) No, obviously not. There may, indeed, be no practical difference, but no practical difference doesn't mean no real difference.
B) Depends on how you define "free will." I don't see how the two are compatible under libertarian free will, where humans are not effected by deterministic physics. It's also hard to square omniscience in the absence of deterministic physics because how else would a being "know" what was going to happen? However, the two are compatible under compatibilists interpretations of free will where free will is less about physical determinism and more about, as you say, agency free from certain constraints.
C) I've never understood how free will is supposed to "resolve" the problem. Free will is about choices, not desires. As Schopenhauer said, we can do what we will, but we can't will what we will. The real question with the problem of evil is why would a God give us the desire to do evil things to begin with? I've used this example before: if there's an old lady crossing the street, I could have two conflicting wills: help her, or ignore her. Whichever I choose could be said to be free will, and one is certainly more moral than the other, but why would me having a will/desire to push the old lady into traffic increase my free will? The obvious answer is that it doesn't, and yet there are plenty of people out there who have the desire to do great evil to people. It makes no sense why a loving God would give us a will to be malicious, as opposed to having the choice between goodness and indifference. Further, the problem of evil isn't just about human actions, but natural ones; subjects like why children are stricken with horrible, natural diseases. Such things have nothing to do with human will, yet still cause us humans great suffering for no apparent reason.
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Post by general313 on May 3, 2020 22:11:01 GMT
A) No, obviously not. There may, indeed, be no practical difference, but no practical difference doesn't mean no real difference. Actually it is. If the decimal expansion of 1/3 is 0.33333.... then 3*(1/3) is 0.9999999... which is one.
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Post by Eva Yojimbo on May 3, 2020 22:48:44 GMT
A) No, obviously not. There may, indeed, be no practical difference, but no practical difference doesn't mean no real difference. Actually it is. If the decimal expansion of 1/3 is 0.33333.... then 3*(1/3) is 0.9999999... which is one. Good point. I guess for actual repeating infinities of the kind you get with pure numbers that is probably true, but if you took to actually enumerating extremely large numbers, like the number of particles in the universe (which, IIRC, is something like 10^80 in the observable universe), if you imagine all but one you'd get an extremely large repeating ".9999" number that wouldn't actually equal 1.
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Post by Arlon10 on May 4, 2020 1:45:29 GMT
Eva Yojimbo said: [ full text here] < clips >
A) No, obviously not. There may, indeed, ... B) ... how else would a being "know" what was going to happen? C) {Why not have several choices, just no really bad choices? (If I read you correctly)] A) Here's how it works. The project leader or foreman or teacher of a certain class or whoever is in charge says whenever I say this (0.999999999999) I mean this (1) and poof! They are now "equal." It happens more than you might know that various endeavors must choose "working" definitions for practical reasons. It is not "illegal." I think maybe you do know. Our words and symbols mean whatever we choose for them to mean. (All definitions are arbitrary.) B) Good question, maybe it requires a superpower. C) Anther good question. I guess I'll have to get back to that one. Meanwhile maybe it's like a simple on/off switch without a means to determine how various choices change the remote future. Consider the science fiction of time travel and what seemingly insignificant act might change the future in very dramatic ways.
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Post by Eva Yojimbo on May 4, 2020 15:01:18 GMT
Eva Yojimbo said: [ full text here] < clips >
A) No, obviously not. There may, indeed, ... B) ... how else would a being "know" what was going to happen? C) {Why not have several choices, just no really bad choices? (If I read you correctly)] A) Here's how it works. The project leader or foreman or teacher of a certain class or whoever is in charge says whenever I say this (0.999999999999) I mean this (1) and poof! They are now "equal." It happens more than you might know that various endeavors must choose "working" definitions for practical reasons. It is not "illegal." I think maybe you do know. Our words and symbols mean whatever we choose for them to mean. (All definitions are arbitrary.) B) Good question, maybe it requires a superpower. C) Anther good question. I guess I'll have to get back to that one. Meanwhile maybe it's like a simple on/off switch without a means to determine how various choices change the remote future. Consider the science fiction of time travel and what seemingly insignificant act might change the future in very dramatic ways. A) This reminds me of Yudkowsky's parable of bleggs and rubes.B) Even with ontological determinism, knowing everything that's going to happen would still be a superpower; but I'm not sure how it would function without that determinism. C) Mostly I was just saying that we can still have free will without the desire to do evil things. I'm not sure what your "it's" in "it's like a simple on/off switch..." is referring to. I've always loved time travel stories though I think most of them are pretty ridiculous even conceptually, since going back in time would require either the creation of new matter for there to be two "you's," or the moment you go back in time it would have to create a new timeline without you in it; either way I don't see how it would work. Still, lots of great stories can be told with the device! I just recently watched the first season of the new Doctor Who and really enjoyed it, eg.
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Post by Dirty Santa PaulsLaugh on May 4, 2020 15:23:59 GMT
0.99999
You can keep adding nines and it will never reach one.
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Post by Deleted on May 4, 2020 15:42:36 GMT
I've asked an actual expert about the does 1 = 0.999r thing.
Apparently, you can produce equations to support both yes and no.
So, it is observer dependent.
They wanted to explain further, but I told them I didn't care.
So the the answer is both yes and no.
Hope that helps 👍
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Post by Dirty Santa PaulsLaugh on May 4, 2020 16:57:37 GMT
C) The problem of evil
The most burned out of all topics, it has long been resolved by the existence of beings with free choices. It can be obvious in this "fallen" world and might not appear in higher realms as far as we know.
Evil is a value assigned to something/one based on cultural/social/sexual bias. What’s evil for you is not evil for me. Certainly there is good and bad, but even this is subject to bias in a lot of cases.
Like, we can all agree the lead poisoning in Flint, MI water system is bad, but is it evil? And while we can say Hitler was evil, there are people who believe he was not because he was acting in accordance with a value system approved within German society at the time.
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Post by general313 on May 4, 2020 21:08:32 GMT
0.99999 You can keep adding nines and it will never reach one. This is like Zeno's paradox of motion "The paradoxical conclusion then would be that travel over any finite distance can neither be completed nor begun, and so all motion must be an illusion."
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Post by Arlon10 on May 8, 2020 1:55:32 GMT
I've asked an actual expert about the does 1 = 0.999r thing. Apparently, you can produce equations to support both yes and no. So, it is observer dependent. They wanted to explain further, but I told them I didn't care. So the the answer is both yes and no. Hope that helps 👍 In calculus there is the concept of the "limit" which is "approached" by various expressions. For example The sum of 9x10 -n as n goes from 1 to infinity approaches a limit of one.
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Post by general313 on May 8, 2020 17:53:33 GMT
I've asked an actual expert about the does 1 = 0.999r thing. Apparently, you can produce equations to support both yes and no. So, it is observer dependent. They wanted to explain further, but I told them I didn't care. So the the answer is both yes and no. Hope that helps 👍 If we limit the discussion to the mathematics of real numbers, there is only one answer: yes. 0.999...
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Post by drystyx on May 8, 2020 21:49:20 GMT
It was way back in high school Math where it was explained why .9999.... =1.
A. 2/3+1/3 =1. Call this A
B. 2/3 = .666... Call this B
C. 1/3=.333....Call this C
D: .333... + .666...= .999. Call this D
All of these are defined.
Because of B and C, 2/3 and 1/3 are interchangeable with .666... and .333....
Therefore, A can be interchanged with D.
Therefore A can be replaced with .666...+ .333....=1.
Therefore, .9999....=1
Now, this is how I was taught. It does seem a bit oxymoronic, and I'm not a fanatic about it, but it's mathematical.
Now, lets get to the fallacy of this principle. The fallacy is in B and C, because we can't fraction out the digits in the base of 10 that we use, so this isn't really definable.
In order to get a defined answer, we need to use a base of "9" instead of 10, in which case we may see the true answer. However, we'd be best to use a base of something like "18", since .9999... would not be usable.
Then, we may see that there is a fallacy in .999...=1. But there isn't, IMO, because we'd see fractions that are different in the base. In a base of 18, it would be different digits. If one wants to work this out, let me know how it turns out. I'm too lazy.
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Post by Arlon10 on May 8, 2020 23:23:13 GMT
I've asked an actual expert about the does 1 = 0.999r thing. Apparently, you can produce equations to support both yes and no. So, it is observer dependent. They wanted to explain further, but I told them I didn't care. So the the answer is both yes and no. Hope that helps 👍 If we limit the discussion to the mathematics of real numbers, there is only one answer: yes. 0.999... Neither infinity, minus infinity, infinitely large, nor infinitesimal are quantities. "1" is a quantity. Just saying.
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Post by general313 on May 8, 2020 23:41:21 GMT
If we limit the discussion to the mathematics of real numbers, there is only one answer: yes. 0.999... Neither infinity, minus infinity, infinitely large, nor infinitesimal are quantities. "1" is a quantity. Just saying. But the sum of an infinite series can be a finite number if it's convergent. For example, .9+.09+.009+...
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Post by Arlon10 on May 9, 2020 0:24:31 GMT
Neither infinity, minus infinity, infinitely large, nor infinitesimal are quantities. "1" is a quantity. Just saying. But the sum of an infinite series can be a finite number if it's convergent. For example, .9+.09+.009+... 1 is the "limit" that the infinite series you mentioned "approaches." The series never reaches that limit. When you speak of the condition "in infinity" it is important to remember there is no such place. Infinity never becomes a quantity in any place. It is best to consider which are and which are not quantities. Infinity is better understood as an "ongoing process" than a quantity. While the assumption of a quantity is not particularly hazardous in this specific instance, it is generally wise to avoid algebraic manipulations involving expressions that are not quantities. Since the use of digital calculators became commonplace it is not unusual to find endeavors that "equate" any digital representative with more accurate notation arbitrarily. Casio has some interesting and inexpensive calculators that can use various modes of expressions.
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Post by general313 on May 9, 2020 18:23:20 GMT
But the sum of an infinite series can be a finite number if it's convergent. For example, .9+.09+.009+... 1 is the "limit" that the infinite series you mentioned "approaches." The series never reaches that limit. When you speak of the condition "in infinity" it is important to remember there is no such place. Infinity never becomes a quantity in any place. It is best to consider which are and which are not quantities. Infinity is better understood as an "ongoing process" than a quantity. While the assumption of a quantity is not particularly hazardous in this specific instance, it is generally wise to avoid algebraic manipulations involving expressions that are not quantities. Since the use of digital calculators became commonplace it is not unusual to find endeavors that "equate" any digital representative with more accurate notation arbitrarily. Casio has some interesting and inexpensive calculators that can use various modes of expressions. If using the idea of "limit" makes you feel more comfortable with the definition, then fine. You can say that the series never reaches that limit if you stop after a finite number of addition steps. Today mathematicians are perfectly comfortable in stating that the "sum" of the infinite series is equal to 1. Interestingly, history is full of examples of discomfort with the idea of certain numbers. In the middle ages and perhaps later, negative numbers were often avoided in mathematical calculations, until it was realized that they were quite handy for intermediate results. The Ancient Greeks were famously uncomfortable with irrational numbers like sqrt(2) because they believed that all numbers should be expressible as a fraction (a ratio of two whole numbers). Even our use of "rational" and "irrational" when applied to thinking reflects this history. This inconvenient truth was apparently so disturbing to the Ancient Greeks that it hampered their development of algebra (they focused on geometry instead).
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Post by Arlon10 on May 9, 2020 19:40:46 GMT
1 is the "limit" that the infinite series you mentioned "approaches." The series never reaches that limit. When you speak of the condition "in infinity" it is important to remember there is no such place. Infinity never becomes a quantity in any place. It is best to consider which are and which are not quantities. Infinity is better understood as an "ongoing process" than a quantity. While the assumption of a quantity is not particularly hazardous in this specific instance, it is generally wise to avoid algebraic manipulations involving expressions that are not quantities. Since the use of digital calculators became commonplace it is not unusual to find endeavors that "equate" any digital representative with more accurate notation arbitrarily. Casio has some interesting and inexpensive calculators that can use various modes of expressions. If using the idea of "limit" makes you feel more comfortable with the definition, then fine. You can say that the series never reaches that limit if you stop after a finite number of addition steps. Today mathematicians are perfectly comfortable in stating that the "sum" of the infinite series is equal to 1.Interestingly, history is full of examples of discomfort with the idea of certain numbers. In the middle ages and perhaps later, negative numbers were often avoided in mathematical calculations, until it was realized that they were quite handy for intermediate results. The Ancient Greeks were famously uncomfortable with irrational numbers like sqrt(2) because they believed that all numbers should be expressible as a fraction (a ratio of two whole numbers). Even our use of "rational" and "irrational" when applied to thinking reflects this history. This inconvenient truth was apparently so disturbing to the Ancient Greeks that it hampered their development of algebra (they focused on geometry instead). That is unless they studied calculus which uses the correct terminology of "limit" I noted earlier. The problem remains which are quantities and which are not. The expression 1/7 is an exact quantity. It's decimal representation is an ongoing division without end. Since it is a repeating chain of digits "142857" the temptation is to assume there is an "arrival" at some quantity once the repeating pattern is known. It can be important to remember that the division never ends though. The expression 1/8 is likewise an exact quantity, however its decimal representation is not ongoing. The division yields an exact quantity of 0.125 because that division is completed at that point. Since calculators often use 8 to 15 digits and truncate the rest, the error caused by decimal representations that are not quantities are not very far off the mark. Remember the radius of a proton. The errors can accumulate in extended calculations though. Notice that even though 1/7 can be expressed as a fraction of integers and its decimal representation is periodic, it will still accumulate errors in extended calculations that depend on decimal representations.
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Post by general313 on May 9, 2020 20:18:48 GMT
If using the idea of "limit" makes you feel more comfortable with the definition, then fine. You can say that the series never reaches that limit if you stop after a finite number of addition steps. Today mathematicians are perfectly comfortable in stating that the "sum" of the infinite series is equal to 1.Interestingly, history is full of examples of discomfort with the idea of certain numbers. In the middle ages and perhaps later, negative numbers were often avoided in mathematical calculations, until it was realized that they were quite handy for intermediate results. The Ancient Greeks were famously uncomfortable with irrational numbers like sqrt(2) because they believed that all numbers should be expressible as a fraction (a ratio of two whole numbers). Even our use of "rational" and "irrational" when applied to thinking reflects this history. This inconvenient truth was apparently so disturbing to the Ancient Greeks that it hampered their development of algebra (they focused on geometry instead). That is unless they studied calculus which uses the correct terminology of "limit" I noted earlier. The problem remains which are quantities and which are not. The expression 1/7 is an exact quantity. It's decimal representation is an ongoing division without end. Since it is a repeating chain of digits "142857" the temptation is to assume there is an "arrival" at some quantity once the repeating pattern is known. It can be important to remember that the division never ends though. The expression 1/8 is likewise an exact quantity, however its decimal representation is not ongoing. The division yields an exact quantity of 0.125 because that division is completed at that point. Since calculators often use 8 to 15 digits and truncate the rest, the error caused by decimal representations that are not quantities are not very far off the mark. Remember the radius of a proton. The errors can accumulate in extended calculations though. Notice that even though 1/7 can be expressed as a fraction of integers and its decimal representation is periodic, it will still accumulate errors in extended calculations that depend on decimal representations. It doesn't matter whether they studied calculus, and I very much doubt that anyone can find a mathematician who hasn't. Since you mentioned the decimal representation of 1/7, it should be pointed out that that representation is exact. Hence one can use the equation 1/7 = 0.142857142857... One does not need to use the approximately equal symbol here (≈).
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Post by Arlon10 on May 9, 2020 23:18:18 GMT
That is unless they studied calculus which uses the correct terminology of "limit" I noted earlier. The problem remains which are quantities and which are not. The expression 1/7 is an exact quantity. It's decimal representation is an ongoing division without end. Since it is a repeating chain of digits "142857" the temptation is to assume there is an "arrival" at some quantity once the repeating pattern is known. It can be important to remember that the division never ends though. The expression 1/8 is likewise an exact quantity, however its decimal representation is not ongoing. The division yields an exact quantity of 0.125 because that division is completed at that point. Since calculators often use 8 to 15 digits and truncate the rest, the error caused by decimal representations that are not quantities are not very far off the mark. Remember the radius of a proton. The errors can accumulate in extended calculations though. Notice that even though 1/7 can be expressed as a fraction of integers and its decimal representation is periodic, it will still accumulate errors in extended calculations that depend on decimal representations. It doesn't matter whether they studied calculus, and I very much doubt that anyone can find a mathematician who hasn't. Since you mentioned the decimal representation of 1/7, it should be pointed out that that representation is exact. Hence one can use the equation 1/7 = 0.142857142857... One does not need to use the approximately equal symbol here (≈). First of all no, many decimal representations are not exact, including that of 1/7. Secondly. even if they were exact, no one has that many digits. Most Calculators do not store more than 15 digits, less expensive ones only 8 digits. Microsoft Excel only uses as many as 15 digits. I wrote a program that can do 100 digit calculations, but I knew full well at the time it is only good for philosophical investigations into the properties of numbers. There is no real world measurement that precise. You are talking about even larger numbers of digits. No matter how small you write you can't get that many digits on a ream of paper. Even Watson the supercomputer can't work with that many digits. It then becomes irrelevant what that many digits might "equal," does it not? In any actual use of mathematics for real world computations it is not a matter of exactitude, it is a matter of necessary precision, for example ±10 -8 is beyond anything you will likely need. You may of course use any notation you wish as long as you and your associates understand each other.
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