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Bet You Can't Define "Truth"
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Originally posted by BlackGriffen: Truth - that which is, irrespective of belief or disbelief of the observer. BlackGriffen |
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Originally posted by Stradlater: Is math ambiguous? |
Well, that question sure is.
But seriously, certain areas of math are quite ambiguous. The Banach-Tarski paradox describes how, using current mathematical practices, one can prove that it is possible to take a sphere with volume x, cut it up into a finite number of pieces, and reassemble those pieces into two spheres, both with a volume of x.
Right... What kind of "truth" is that?
To be fair, though, when I made my comment about language being ambiguous, I wasn't really referring to the language of mathematics.
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Originally posted by Turias: Well, that question sure is. But seriously, certain areas of math are quite ambiguous. The Banach-Tarski paradox describes how, using current mathematical practices, one can prove that it is possible to take a sphere with volume x, cut it up into a finite number of pieces, and reassemble it into two spheres, both with a volume of x. Right... What kind of "truth" is that? To be fair, though, when I made my comment about language being ambiguous, I wasn't really referring about the language of mathematics. |
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Originally posted by Stradlater: I'm not familiar with the banach-tarski paradox, strange (link please?)...but this reminds me of the simple proof in which 2=1, but the problem was that there was a divide-by-zero, I wonder if certain mathematical rules are broken in the example you mention, but perhaps not. |
No, in the aformentioned paradox, the "trick" lies in the usage of a certain step that has been used to "prove" many other mathematical ideas that most mathematicians hold to be "true". We have not yet been able to prove many of these ideas without that step, and people continue to use it as if it were "true". Banach and Tarski were trying to show the mathematic community just how ridiculous they were being by continuing to use this step in their proofs. However, it didn't stop them.
As far as linking the paradox, I haven't found the ideal link, so I'll just point you to Google. There are a ton of pages out there on the subject as it is a fairly popular paradox.
Interesting. In the paradox, it looks like the "Axiom of Choice" isn't known to be absolutely true (or at least certain exceptions remain to be absolutely defined).
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Originally posted by Stradlater: Interesting. In the paradox, it looks like the "Axiom of Choice" isn't known to be absolutely true (or at least certain exceptions remain to be absolutely defined). |
Exactly, which just goes to show how even mathematics and "truth" can not go hand-in-hand. For how can an axiom (a self-evident or universally recognized truth) not be true?
Well, infinite divisibility can make strange things occur that can't happen in the "real world". An atom, for example, can be moved, but being sliced into small pieces and moved about how we like isn't a feature I know of.
You can get other rather unintuitive ideas from fractals, for example the Cantor middle-thirds set. http://www.math.buffalo.edu/~sww/cl...T/COMPACT2.html
If you ignore the compactness stuff for a moment, essentially you have a subset of [0,1] that has measure 0 but there exists a bijection from it to [0,1] i.e. it has length 1! That seems highly unintuitive, and in fact it should be impossible to construct such a set in the "real world" because matter is not infinitely divisible... at least not smoothly.
Matt Fahrenbacher
You can get other rather unintuitive ideas from fractals, for example the Cantor middle-thirds set. http://www.math.buffalo.edu/~sww/cl...T/COMPACT2.html
If you ignore the compactness stuff for a moment, essentially you have a subset of [0,1] that has measure 0 but there exists a bijection from it to [0,1] i.e. it has length 1! That seems highly unintuitive, and in fact it should be impossible to construct such a set in the "real world" because matter is not infinitely divisible... at least not smoothly.
Matt Fahrenbacher
Well, an Axiom (a given) is always true. The question is if under the mathematical system they have developed, are they assuming that the axiom of choice is really true? Depending on what axioms you hold to be true to begin with, axioms that are held to be true in different mathematicaly systems need not hold.
Matt Fahrenbacher
Matt Fahrenbacher
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Originally posted by Ghoser777: Well, infinite divisibility... |
1. I'm pretty sure it was FINITE divisibility.
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| Well, an Axiom (a given) is always true. The question is if under the mathematical system they have developed, are they assuming that the axiom of truth is really true? Depending on what axioms you hold to be true to begin with, axioms that are held to be true in different mathematicaly systems need not hold. |
2. Axiom of Choice
1. I'm pretty sure it was FINITE divisibility.
Actually it was infinite divisibility. Sure, you were breaking up the circle into 5 (or four) sections, but the edges weren't exactly your everyday cuts. They were ragged, and probably impossible in terms of real world matter.
2. Axiom of Choice
beh, typed too fast, you know what I meant I'm just not sure they assumed the axiom of choice to be true when presenting the problem. If the axiom of choice turns out to not be true in the given situation, then the paradox is invalid in any system where the axiom of choice is valid.
Matt Fahrenbacher
Actually it was infinite divisibility. Sure, you were breaking up the circle into 5 (or four) sections, but the edges weren't exactly your everyday cuts. They were ragged, and probably impossible in terms of real world matter.
2. Axiom of Choice
beh, typed too fast, you know what I meant
I'm just not sure they assumed the axiom of choice to be true when presenting the problem. If the axiom of choice turns out to not be true in the given situation, then the paradox is invalid in any system where the axiom of choice is valid.Matt Fahrenbacher
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Originally posted by Ghoser777: I'm just not sure they assumed the axiom of choice to be true when presenting the problem. If the axiom of choice turns out to not be true in the given situation, then the paradox is invalid in any system where the axiom of choice is valid. |
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Originally posted by Ghoser777: Actually it was infinite divisibility. Sure, you were breaking up the circle into 5 (or four) sections, but the edges weren't exactly your everyday cuts. They were ragged, and probably impossible in terms of real world matter. |
Of course it is impossible in terms of real world matter. That's not the point. The point is that given our system of mathematics and the axioms we hold "true", it is possible.
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| I'm just not sure they assumed the axiom of choice to be true when presenting the problem. |
They did assume the Axiom of Choice to be true. They were then (basically) attempting to disprove it by contradiction.
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| If the axiom of choice turns out to not be true in the given situation, then the paradox is invalid in any system where the axiom of choice is valid. |
I'm not really sure what you mean by "given situation". An axiom, by definition, is always true. If it isn't, it isn't an axiom. If the Axiom of Choice turns out to not really be an axiom, then we are going to have to go back and re-prove a lot of things...
Gah, I had a response, and then iTunes and safari started freaking out together and hogging as much processor as they could 
Of course it is impossible in terms of real world matter. That's not the point. The point is that given our system of mathematics and the axioms we hold "true", it is possible.
Sure, but you get all kinds of wacky things that don't make any sense in the real world with infinite divisibility - the only reason they seem wacky is because in the real world you don't have infinite divisibility for matter (although you do for time and such, or so we think). Case in point was the set with length 1 but measure 0 - the Cantor middle third set.
What this paradox shows us is that with infinite divisibility, area and volume and the like are not necessarily as nicely defined and easy to think about as in the "real world." It only seems paradoxical from an intuitive stand point, because it does seem to imply that 2 equals 1, but the cantor set seems to imply that 1 equals 0 and that's not really paradoxical - it just assumes infinite divisibility.
They did assume the Axiom of Choice to be true. They were then (basically) attempting to disprove it by contradiction.
Sounds good to me.
I'm not really sure what you mean by "given situation". An axiom, by definition, is always true. If it isn't, it isn't an axiom. If the Axiom of Choice turns out to not really be an axiom, then we are going to have to go back and re-prove a lot of things...
In any given situation, you can assume whatever axioms you want to be true. Like you can assume that the shortest distance between two points is a circle - then instead of talking about Euclidean geometry, you're talking about Spherical geometry. Essentially, based on what axioms you assume to be true, you get different results. So, all I was saying is an axiom is only true if we assume it to be so. There's plenty of different mathematical structures that are created where certain field, group, etc axioms are assumed to be true or false.
Matt Fahrenbacher
Of course it is impossible in terms of real world matter. That's not the point. The point is that given our system of mathematics and the axioms we hold "true", it is possible.
Sure, but you get all kinds of wacky things that don't make any sense in the real world with infinite divisibility - the only reason they seem wacky is because in the real world you don't have infinite divisibility for matter (although you do for time and such, or so we think). Case in point was the set with length 1 but measure 0 - the Cantor middle third set.
What this paradox shows us is that with infinite divisibility, area and volume and the like are not necessarily as nicely defined and easy to think about as in the "real world." It only seems paradoxical from an intuitive stand point, because it does seem to imply that 2 equals 1, but the cantor set seems to imply that 1 equals 0 and that's not really paradoxical - it just assumes infinite divisibility.
They did assume the Axiom of Choice to be true. They were then (basically) attempting to disprove it by contradiction.
Sounds good to me.
I'm not really sure what you mean by "given situation". An axiom, by definition, is always true. If it isn't, it isn't an axiom. If the Axiom of Choice turns out to not really be an axiom, then we are going to have to go back and re-prove a lot of things...
In any given situation, you can assume whatever axioms you want to be true. Like you can assume that the shortest distance between two points is a circle - then instead of talking about Euclidean geometry, you're talking about Spherical geometry. Essentially, based on what axioms you assume to be true, you get different results. So, all I was saying is an axiom is only true if we assume it to be so. There's plenty of different mathematical structures that are created where certain field, group, etc axioms are assumed to be true or false.
Matt Fahrenbacher
Truth is what you believe in.
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Originally posted by Ghoser777: Sure, but you get all kinds of wacky things that don't make any sense in the real world with infinite divisibility - the only reason they seem wacky is because in the real world you don't have infinite divisibility for matter (although you do for time and such, or so we think). Case in point was the set with length 1 but measure 0 - the Cantor middle third set. What this paradox shows us is that with infinite divisibility, area and volume and the like are not necessarily as nicely defined and easy to think about as in the "real world." It only seems paradoxical from an intuitive stand point, because it does seem to imply that 2 equals 1, but the cantor set seems to imply that 1 equals 0 and that's not really paradoxical - it just assumes infinite divisibility. |
I agree that there are a ton of strange things you can do with infinity. There's the Cantor set, the cone which can hold a finite amount of paint but would take an infinite amount of paint to cover, etc, etc. But these are in a completely different set than the Banach-Tarski paradox.
The thing is, the Banach-Tarski paradox doesn't involve infinity and lives in a completely different realm of mathematics. Plus, it's a paradox meant to prove something false, unlike the others.
The ones you mention are indeed interesting, but they are fundamentally different than the B-T.
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| In any given situation, you can assume whatever axioms you want to be true. Like you can assume that the shortest distance between two points is a circle - then instead of talking about Euclidean geometry, you're talking about Spherical geometry. Essentially, based on what axioms you assume to be true, you get different results. So, all I was saying is an axiom is only true if we assume it to be so. There's plenty of different mathematical structures that are created where certain field, group, etc axioms are assumed to be true or false. |
Don't you mean "arc" instead of "circle"? Anyway, what you say is not entirely true. I think you are talking about transcending different coordinate systems and realities. The B-T paradox does neither. An axiom is an axiom and is always true. If you are talking about coordinate system A, then the axiom must say as much. An axiom can't be true in some situations and false in others. The B-T paradox is the same. It can't be true sometimes and false other times. It either is, or isn't.
I think we've done a pretty good job of derailing this thread from "truth" to mathematics. It's all your fault, Stradlater!
I agree that there are a ton of strange things you can do with infinity. There's the Cantor set, the cone which can hold a finite amount of paint but would take an infinite amount of paint to cover, etc, etc. But these are in a completely different set than the Banach-Tarski paradox.
Are you so sure?
From http://www.kuro5hin.org/story/2003/5/23/134430/275
The thing is, the Banach-Tarski paradox doesn't involve infinity and lives in a completely different realm of mathematics. Plus, it's a paradox meant to prove something false, unlike the others.
The ones you mention are indeed interesting, but they are fundamentally different than the B-T.
They might be fundamentally different, I do not know. But here is another quote from the article that I think helps my point:
I really think this article is a good read
Don't you mean "arc" instead of "circle"? Anyway, what you say is not entirely true. I think you are talking about transcending different coordinate systems and realities. The B-T paradox does neither. An axiom is an axiom and is always true. If you are talking about coordinate system A, then the axiom must say as much. An axiom can't be true in some situations and false in others. The B-T paradox is the same. It can't be true sometimes and false other times. It either is, or isn't.
"Great Circle" is the technical term for a straight line on a sphere. Considering we're not really talking about the Sphere in free space, but instead as THE space, a circle is an okay word to use.
An axion is not always true. An axiom is only true if you assume it to be true. Here's an axiom:
"All men are pigs"
This axiom could be true in general, I don't know. But we only know it to be true if we assume it to be so.
Now. once you've assumed your axiom, it can only be false if you're mathematicaly system is inconsitent (you've made two contrary assumptions, something that happens a lot in proofs by contradiction).
You can develop different mathematical systems (groups, fields, etc) based on what axioms you assume to be true. The common field axioms doesn't apply to the set of all 3X3 matricies (no multiplicative commutitivity) for example.
Matt Fahrenbacher
Are you so sure?
From http://www.kuro5hin.org/story/2003/5/23/134430/275
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| In fact, if we assume that spheres are not infinitely divisible, then the Banach-Tarski paradox doesn't apply, because each of the "pieces" in the paradox is so infinitely complex that they are not "measurable" (in human language, they do not have a well-defined volume; it is impossible to measure their volume). Immeasurable pieces can only exist if the sphere can be cut into infinitely-detailed pieces; this obviously isn't true for real spheres, since you cannot cut atoms into arbitrary shapes, especially not into infinitely complex shapes. |
The thing is, the Banach-Tarski paradox doesn't involve infinity and lives in a completely different realm of mathematics. Plus, it's a paradox meant to prove something false, unlike the others.
The ones you mention are indeed interesting, but they are fundamentally different than the B-T.
They might be fundamentally different, I do not know. But here is another quote from the article that I think helps my point:
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| (Before you dismiss this notion outright, let me state that mathematically infinite objects do not always behave intuitively. As a comparison, we use a more intuitive example of duplicating the set of integers: given N, the (infinitely large) set containing all the integers, we can split them up into two sets, E containing all the even integers, and F containing all the odd integers. Are E and F each smaller than N, the set of all integers? Intuitively, it appears to be so; however, I will convince you that they are, in fact, the same size. First, we take E, and rename each member of E so that a number x is renamed to x divided by two. What do we get? We now find that E=N. Similarly, we take each member y from F, and rename y to (y-1)/2. Whoopie, we also find that F=N. We have just duplicated the set of integers using nothing more than just the original integers. We didn't even need to use infinitely-divisible freak objects to achieve this.) |
I really think this article is a good read
Don't you mean "arc" instead of "circle"? Anyway, what you say is not entirely true. I think you are talking about transcending different coordinate systems and realities. The B-T paradox does neither. An axiom is an axiom and is always true. If you are talking about coordinate system A, then the axiom must say as much. An axiom can't be true in some situations and false in others. The B-T paradox is the same. It can't be true sometimes and false other times. It either is, or isn't.
"Great Circle" is the technical term for a straight line on a sphere. Considering we're not really talking about the Sphere in free space, but instead as THE space, a circle is an okay word to use.
An axion is not always true. An axiom is only true if you assume it to be true. Here's an axiom:
"All men are pigs"
This axiom could be true in general, I don't know. But we only know it to be true if we assume it to be so.
Now. once you've assumed your axiom, it can only be false if you're mathematicaly system is inconsitent (you've made two contrary assumptions, something that happens a lot in proofs by contradiction).
You can develop different mathematical systems (groups, fields, etc) based on what axioms you assume to be true. The common field axioms doesn't apply to the set of all 3X3 matricies (no multiplicative commutitivity) for example.
Matt Fahrenbacher
Ok, What is "Truth"
... whatever I agree with.
... whatever I agree with.
Defining 'Truth' is easy. Identifying it is another matter entirely.
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Originally posted by Turias: It's all your fault, Stradlater!
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Of course there is "Truth" (or absolute truth). Whether or not anybody ever sees or understands it is doubtful.
We all have our perspectives of the "Truth". Something must exist to have a perspective of it, correct? So doesn't it follow that there must be a Truth for us to have a perspective of?
We all have our perspectives of the "Truth". Something must exist to have a perspective of it, correct? So doesn't it follow that there must be a Truth for us to have a perspective of?