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Topic: infinities and the mind-dependence of maths (split from Newton etc.) (Read 1109 times) previous topic - next topic - Topic derived from Newton, Copernicus, G...

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  • Brother Daniel
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Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #25
russell's paradox says that you can't take a set of all sets because you would have to include that set in the set of all sets which puts you in an infinite recursive (procedural) loop of operations. Disregarding classes and other ways to avoid the paradox, it is a paradox because the 'set of all sets' includes the results of an iterative process.  Each iteration satisfies it's mandate but needs to be included in following iterations. It is only a paradox if we consider the set of all sets to be outside the iterative process.
No, there's nothing inherently paradoxical about self-referentiality.  Nor am I seeing the infinite recursive procedural loop.  Iterative processes can be useful in finding solutions to some problems, but there's nothing about self-referentiality that demands them.

Russell's Paradox is specifically about the set of all sets that lack the property of being an element of itself.  The existence of that set is self-contradictory.  This has nothing to do with any "iterative processes".  (And if we put suitable restrictions on what we consider to be a "set" (e.g. using ZF) then the paradox disappears.)

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #26
russell's paradox says that you can't take a set of all sets because you would have to include that set in the set of all sets which puts you in an infinite recursive (procedural) loop of operations. Disregarding classes and other ways to avoid the paradox, it is a paradox because the 'set of all sets' includes the results of an iterative process.  Each iteration satisfies it's mandate but needs to be included in following iterations. It is only a paradox if we consider the set of all sets to be outside the iterative process.
No, there's nothing inherently paradoxical about self-referentiality.  Nor am I seeing the infinite recursive procedural loop.  Iterative processes can be useful in finding solutions to some problems, but there's nothing about self-referentiality that demands them.

Russell's Paradox is specifically about the set of all sets that lack the property of being an element of itself.  The existence of that set is self-contradictory.  This has nothing to do with any "iterative processes".  (And if we put suitable restrictions on what we consider to be a "set" (e.g. using ZF) then the paradox disappears.)
The existence of that set is only an issue because of the iterative nature of the issue. When you make that set, it IS the set of all sets. It only fails to be if you ask the question again after you make the set. Not before. The problem is the belief in a god's eye view that there is no before and after.

ETA: The question asks for a literally different thing after each operation.
Love is like a magic penny
 if you hold it tight you won't have any
if you give it away you'll have so many
they'll be rolling all over the floor

  • Brother Daniel
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Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #27
A set of {x} is an operation. No?
I don't see sets as operations, no.  Seems like a category error.
Quote from: Testy
ETA: or are you saying this isn't a case where an excluded middle applies?
No, not saying that.  (No idea where you got this.)

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #28
Love is like a magic penny
 if you hold it tight you won't have any
if you give it away you'll have so many
they'll be rolling all over the floor

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #29
A set of {x} is an operation. No?
I don't see sets as operations, no.  Seems like a category error.
How could it not be an operation when you actually have to define the contents of a set?

Quote
Quote from: Testy
ETA: or are you saying this isn't a case where an excluded middle applies?
No, not saying that.  (No idea where you got this.)

I was grasping at straws.
Love is like a magic penny
 if you hold it tight you won't have any
if you give it away you'll have so many
they'll be rolling all over the floor

  • Brother Daniel
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Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #30
russell's paradox says that you can't take a set of all sets because you would have to include that set in the set of all sets which puts you in an infinite recursive (procedural) loop of operations. Disregarding classes and other ways to avoid the paradox, it is a paradox because the 'set of all sets' includes the results of an iterative process.  Each iteration satisfies it's mandate but needs to be included in following iterations. It is only a paradox if we consider the set of all sets to be outside the iterative process.
No, there's nothing inherently paradoxical about self-referentiality.  Nor am I seeing the infinite recursive procedural loop.  Iterative processes can be useful in finding solutions to some problems, but there's nothing about self-referentiality that demands them.

Russell's Paradox is specifically about the set of all sets that lack the property of being an element of itself.  The existence of that set is self-contradictory.  This has nothing to do with any "iterative processes".  (And if we put suitable restrictions on what we consider to be a "set" (e.g. using ZF) then the paradox disappears.)
The existence of that set is only an issue because of the iterative nature of the issue.
Again, I'm not seeing any "iterative nature" here.
Quote from: Testy
When you make that set, it IS the set of all sets.
I wasn't talking about the set of all sets.
Quote from: Testy
It only fails to be if you ask the question again after you make the set. Not before.
Which question?  The question of whether that (alleged) set is an element of itself?
Quote from: Testy
The problem is the belief in a god's eye view that there is no before and after.
There is no before and after.

You can call that a "god's eye view" if you like, but I think that's just a subtle poisoning of the well.

  • Brother Daniel
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Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #31
A set of {x} is an operation. No?
I don't see sets as operations, no.  Seems like a category error.
How could it not be an operation when you actually have to define the contents of a set?
How does the former follow in any way from the latter?

(And what point were you making when you first said "a set of {x} is an operation"?  I don't see how that addresses anything in the post of mine to which you were responding.)

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #32

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #33
russell's paradox says that you can't take a set of all sets because you would have to include that set in the set of all sets which puts you in an infinite recursive (procedural) loop of operations. Disregarding classes and other ways to avoid the paradox, it is a paradox because the 'set of all sets' includes the results of an iterative process.  Each iteration satisfies it's mandate but needs to be included in following iterations. It is only a paradox if we consider the set of all sets to be outside the iterative process.
No, there's nothing inherently paradoxical about self-referentiality.  Nor am I seeing the infinite recursive procedural loop.  Iterative processes can be useful in finding solutions to some problems, but there's nothing about self-referentiality that demands them.

Russell's Paradox is specifically about the set of all sets that lack the property of being an element of itself.  The existence of that set is self-contradictory.  This has nothing to do with any "iterative processes".  (And if we put suitable restrictions on what we consider to be a "set" (e.g. using ZF) then the paradox disappears.)
The existence of that set is only an issue because of the iterative nature of the issue.
Again, I'm not seeing any "iterative nature" here.
Quote from: Testy
When you make that set, it IS the set of all sets.
I wasn't talking about the set of all sets.
Quote from: Testy
It only fails to be if you ask the question again after you make the set. Not before.
Which question?  The question of whether that (alleged) set is an element of itself?
Yes. It can't be until it exists. Once it exists, the question then needs to be asked again.

Here, let's go ahead and use a more formalized formulation from wiki:
Now we consider the set of all normal sets, R. Determining whether R is normal or abnormal is impossible: if R were a normal set, it would be contained in the set of normal sets (itself), and therefore be abnormal; and if R were abnormal, it would not be contained in the set of all normal sets (itself), and therefore be normal. This leads to the conclusion that R is neither normal nor abnormal: Russell's paradox.

The set R cannot be analyzed until it is made. Once it is made, a new set R(1) needs to be made to satisfy the NEW question, is the set of all sets a member of itself? [ETA:The set is no longer the set of all sets.] The paradox that such a set is neither normal nor abnormal is only a paradox of sleight of hand regarding timing. It is identical to asking if a person is alive or dead when you remove time from the equation.
Quote
Quote from: Testy
The problem is the belief in a god's eye view that there is no before and after.
There is no before and after.

You can call that a "god's eye view" if you like, but I think that's just a subtle poisoning of the well.
A countable infinity means only that no clear impediment to the process of counting can be seen.
  • Last Edit: September 07, 2016, 12:21:01 PM by Testy Calibrate
Love is like a magic penny
 if you hold it tight you won't have any
if you give it away you'll have so many
they'll be rolling all over the floor

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #34
Love is like a magic penny
 if you hold it tight you won't have any
if you give it away you'll have so many
they'll be rolling all over the floor

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #35
Happy for you

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #36
Quote from: Testy
The problem is the belief in a god's eye view that there is no before and after.
There is no before and after.

to an operation?
Love is like a magic penny
 if you hold it tight you won't have any
if you give it away you'll have so many
they'll be rolling all over the floor

  • Brother Daniel
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Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #37
Yes. It can't be until it exists.
What is "it" here?  I can't parse this.
Quote from: Testy
Once it exists, the question then needs to be asked again.
"Again" seems to contradict your previous sentence.
Quote from: Testy
The set R cannot be analyzed until it is made. Once it is made, a new set R(1) needs to be made to satisfy the NEW question, is the set of all sets a member of itself? [ETA:The set is no longer the set of all sets.]
I can't follow you at all.  You appear to be mixing several different concepts together.
Quote from: Testy
The paradox that such a set is neither normal nor abnormal is only a paradox of sleight of hand regarding timing. It is identical to asking if a person is alive or dead when you remove time from the equation.
What.  The.  Hell.  Are.  You.  Smoking.
Quote from: Testy
A countable infinity means only that no clear impediment to the process of counting can be seen.
No, it means that there's a one-to-one correspondence between the set of natural numbers and the set in question.

  • Pingu
Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #38
Thread is now about infinities.
told ya
Talk about a wank fest

Not really.  Actual math turns out to be useful.  Numberwanking isn't.

  • Brother Daniel
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Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #39
Quote from: Testy
The problem is the belief in a god's eye view that there is no before and after.
There is no before and after.
to an operation?
To a mathematical operation.

Consider the operation of adding 2 and 3.

2 + 3 = 5.  It just is.  It's timelessly true.

It's rubbish to say "well, we don't have 5 until we execute the operation of adding 2 and 3."  A mathematical operation is not something that has to be "executed" as such; it's not embedded in time.

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #40
Yes. It can't be until it exists.
What is "it" here?  I can't parse this.
Quote from: Testy
Once it exists, the question then needs to be asked again.
"Again" seems to contradict your previous sentence.
Quote from: Testy
The set R cannot be analyzed until it is made. Once it is made, a new set R(1) needs to be made to satisfy the NEW question, is the set of all sets a member of itself? [ETA:The set is no longer the set of all sets.]
I can't follow you at all.  You appear to be mixing several different concepts together.

I am saying that there is no way to ever make that set R removed from time. Once you make it, it is no longer R. However, as an answer to the operation, 'the set of all sets', it was a legitimate answer to the operation at the time it was asked. It will not satisfy the operation again though since a new set would need to include it. Then, were the operation to be run again, neither that new set nor the original set would satisfy the operation but a new set would. It is a simple error but assaults something fairly sacred in maths. Not sure exactly why. I doubt mathematicians would have a problem calling each successive rubber ducky rolling off an assembly line the 'newest' rubber ducky and yet they have exactly that problem with the set of all sets.
Quote from: Testy
The paradox that such a set is neither normal nor abnormal is only a paradox of sleight of hand regarding timing. It is identical to asking if a person is alive or dead when you remove time from the equation.
What.  The.  Hell.  Are.  You.  Smoking.[/quote] Not the best example. See the rubber ducky above.
Quote
Quote from: Testy
A countable infinity means only that no clear impediment to the process of counting can be seen.
No, it means that there's a one-to-one correspondence between the set of natural numbers and the set in question.

No. It means no clear impediment to making a one-to-one correspondence between elements can be imagined. Your answer is exactly wrong. That is not only unknown but unknowable. All that can be known about any continuum is that no obvious impediment to its continuation can be imagined. There is always a horizon. Always. Finite beings in an infinite universe. In fact, horizons are our only indication of infinity.
Love is like a magic penny
 if you hold it tight you won't have any
if you give it away you'll have so many
they'll be rolling all over the floor

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #41
Quote from: Testy
The problem is the belief in a god's eye view that there is no before and after.
There is no before and after.
to an operation?
To a mathematical operation.

Consider the operation of adding 2 and 3.

2 + 3 = 5.  It just is.  It's timelessly true.

It's rubbish to say "well, we don't have 5 until we execute the operation of adding 2 and 3."  A mathematical operation is not something that has to be "executed" as such; it's not embedded in time.

I disagree emphatically and profoundly. 2+3=5 is exactly a perfect example. It doesn't equal anything at all without a mind. It doesn't even have meaning without something to parse it and that parsing is individual each and every time it is done. You may remember the answer but the answer is the result of the operation. That it is the same every time is irrelevant. You are arguing for god pure and simple. Either it happens individually in minds or it is held in some uber mind but nowhere in the universe of things is it held.
Love is like a magic penny
 if you hold it tight you won't have any
if you give it away you'll have so many
they'll be rolling all over the floor

  • Brother Daniel
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Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #42
I disagree emphatically and profoundly. 2+3=5 is exactly a perfect example. It doesn't equal anything at all without a mind. It doesn't even have meaning without something to parse it and that parsing is individual each and every time it is done.
"It doesn't even have meaning without something to parse it" is true provided that by "it" we're referring to the string of symbols that appears as "2 + 3 = 5".  But the symbols are not the same as the underlying reality.  I think you're making a map-territory error again.
Quote from: Testy
You are arguing for god pure and simple.
I don't think I am.  But I'm not much bothered if you want to label me a theist.  There are worse things to be.
Quote from: Testy
Either it happens individually in minds or it is held in some uber mind but nowhere in the universe of things is it held.
I'm not surprised that you don't see the "universe of things" as including mathematical truths in themselves, but I am surprised that you don't see the "universe of things" as including minds.  If this stuff takes place in minds, then surely it is held in "the universe of things".

In any case, the mind-dependence (of sets, numbers, etc.) seems to be what we're disputing.

Gotta run.  I'll get back to this later.

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #43
Maybe you guys should get a room

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #44
I disagree emphatically and profoundly. 2+3=5 is exactly a perfect example. It doesn't equal anything at all without a mind. It doesn't even have meaning without something to parse it and that parsing is individual each and every time it is done.
"It doesn't even have meaning without something to parse it" is true provided that by "it" we're referring to the string of symbols that appears as "2 + 3 = 5".  But the symbols are not the same as the underlying reality.  I think you're making a map-territory error again.
What could we possibly be referring to other than a statement that "the operation of taking 2 and adding 3 produces the result of 5"? What territory is this mapping? I am especially interested in your answer to this because I too think there is an underlying map territory issue going on but I don't see it the same as you do. Like I said, I am a terrible platonist as well as a terrible materialist. I am a here and nowist I suppose. Or maybe a buddhist flavor of some sort. At any rate, you seem to be saying that because an operation produces the same result every time due to the logic of its parameters, that this result exists independently of the operation. I can't even see how that is a sensical statement if that is what you are saying,

Quote
Quote from: Testy
You are arguing for god pure and simple.
I don't think I am.  But I'm not much bothered if you want to label me a theist.  There are worse things to be.
I am not bothered by it either but it does lead to the qualities of the deity in question. You are making a strong deterministic statement by allowing that particular type of god.

Quote
Quote from: Testy
Either it happens individually in minds or it is held in some uber mind but nowhere in the universe of things is it held.
I'm not surprised that you don't see the "universe of things" as including mathematical truths in themselves, but I am surprised that you don't see the "universe of things" as including minds.  If this stuff takes place in minds, then surely it is held in "the universe of things".
Like I said, I'm not a good materialist either. But minds are for making maps of experience and the contents of a mind are not useful or meaningful outside of the operation of that mind, so again, we are back to iterative processes changing the landscape and minds navigating time as if it were malleable thereby applying unwarranted equalities between discrete unequal temporal landscapes.

Quote

In any case, the mind-dependence (of sets, numbers, etc.) seems to be what we're disputing.

Gotta run.  I'll get back to this later.


I think it is, or at least it's involved.
Love is like a magic penny
 if you hold it tight you won't have any
if you give it away you'll have so many
they'll be rolling all over the floor

  • Brother Daniel
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  • predisposed to antagonism
Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #45
I am saying that there is no way to ever make that set R removed from time. Once you make it, it is no longer R. However, as an answer to the operation, 'the set of all sets', it was a legitimate answer to the operation at the time it was asked. It will not satisfy the operation again though since a new set would need to include it. Then, were the operation to be run again, neither that new set nor the original set would satisfy the operation but a new set would.
Again, this is too nonsensical even to be wrong.  I don't know whether the ideas you're trying to get across are themselves nonsensical, or whether it's only your presentation of them that is gibberish.
Quote from: Testy
Not the best example. See the rubber ducky above.
Doesn't help.
Quote from: Testy
Quote from: BD
Quote from: Testy
A countable infinity means only that no clear impediment to the process of counting can be seen.
No, it means that there's a one-to-one correspondence between the set of natural numbers and the set in question.
No. It means no clear impediment to making a one-to-one correspondence between elements can be imagined. Your answer is exactly wrong. That is not only unknown but unknowable.
Rubbish.  You can use a different definition than the mathematicians do, if you want, but then you're talking about a different concept (unless you can prove that the definitions are equivalent).

We can establish that ℤ is countable -- that there really is a one-to-one correspondence between ℕ and ℤ, not merely that "we can't imagine an impediment" -- quite clearly:
Consider ℤ, the set of integers.  We can easily find a bijection f : ℕ → ℤ defined by
f(1) = 0;
f(2n) = n for every n ∈ ℕ;
f(2n + 1) = -n for every n ∈ ℕ.
This is clearly a one-to-one correspondence.  You seem to be saying that because I'm dealing with infinite sets, there must be some possibility that something about this correspondence will break down somewhere.  Infinite sets aren't really that special.

  • JonF
Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #46
Consider ℤ, the set of integers.  We can easily find a bijection f : ℕ → ℤ defined by
f(1) = 0;
f(2n) = n for every n ∈ ℕ;
f(2n + 1) = -n for every n ∈ ℕ.
This is clearly a one-to-one correspondence.  You seem to be saying that because I'm dealing with infinite sets, there must be some possibility that something about this correspondence will break down somewhere.  Infinite sets aren't really that special.
Seen Good Math/Bad Math: Cantor Crankery?
"I would never consider my evaluation of his work to be fair minded unless I had actually read his own words." - Dave Hawkins

  • Brother Daniel
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Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #47
What could we possibly be referring to other than a statement that "the operation of taking 2 and adding 3 produces the result of 5"?
There's the statement, and then there's the underlying truth reflected in the statement.  "Meaning" is the mapping from one to the other.  Of course a "statement" has no meaning without a mind to parse it, but that has no implications about the underlying mathematical truth.
Quote from: Testy
At any rate, you seem to be saying that because an operation produces the same result every time due to the logic of its parameters, that this result exists independently of the operation. I can't even see how that is a sensical statement if that is what you are saying,
I'm not saying that the result exists independently of the operation.  I agree that that would be nonsensical.  I'm saying (among other things) that the operation exists independently of statements of the operation.  And that the "operation" is not best viewed as a physical process that happens in time.

I made no reference to the operation producing the same result every time.  In your narrative, it's not even true that the operation produces the same result every time.  Since you're taking the operation to be nothing but a mental process, the fact is that the mental process of "doing" a mathematical operation can give variable results.  There won't be much variation in 2+3, of course, but there can be a lot of variation when the questions get harder.

It's easy to account for such things in my narrative:  There's a right answer (2+3=5), but sometimes people make errors (2+3=6).  Since you're denying the existence of a mind-independent underlying mathematical truth, I don't see how you can refer to "errors" in your narrative, unless you do something weird like trying to make mathematical truths democratic (e.g. "usually when people contemplate 2+3 they get 5, so we'll take that to be 'right'").  Of course the word "usually" will itself depend on some sort of mathematical understanding as well, so the democracy-as-truth model would probably break down under its own weight.
Quote from: Testy
I am not bothered by it either but it does lead to the qualities of the deity in question. You are making a strong deterministic statement by allowing that particular type of god.
You have to presume what we're disputing (among other things) in order to argue that I'm taking a theistic position at all.  Naturally, I don't accept that.  Now you're intimating that I'm allowing a "particular type" of god, without even hinting at your reasoning.  I'd ask you to show your work, but I'm not sure that I care at this point.
Quote from: Testy
so again, we are back to iterative processes changing the landscape and minds navigating time as if it were malleable thereby applying unwarranted equalities between discrete unequal temporal landscapes.
Did you get this from a random pomo generator?

  • Brother Daniel
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  • predisposed to antagonism
Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #48

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #49
I am saying that there is no way to ever make that set R removed from time. Once you make it, it is no longer R. However, as an answer to the operation, 'the set of all sets', it was a legitimate answer to the operation at the time it was asked. It will not satisfy the operation again though since a new set would need to include it. Then, were the operation to be run again, neither that new set nor the original set would satisfy the operation but a new set would.
Again, this is too nonsensical even to be wrong.  I don't know whether the ideas you're trying to get across are themselves nonsensical, or whether it's only your presentation of them that is gibberish.
Quote from: Testy
Not the best example. See the rubber ducky above.
Doesn't help.
Quote from: Testy
Quote from: BD
Quote from: Testy
A countable infinity means only that no clear impediment to the process of counting can be seen.
No, it means that there's a one-to-one correspondence between the set of natural numbers and the set in question.
No. It means no clear impediment to making a one-to-one correspondence between elements can be imagined. Your answer is exactly wrong. That is not only unknown but unknowable.
Rubbish.  You can use a different definition than the mathematicians do, if you want, but then you're talking about a different concept (unless you can prove that the definitions are equivalent).

We can establish that ℤ is countable -- that there really is a one-to-one correspondence between ℕ and ℤ, not merely that "we can't imagine an impediment" -- quite clearly:
Consider ℤ, the set of integers.  We can easily find a bijection f : ℕ → ℤ defined by
f(1) = 0;
f(2n) = n for every n ∈ ℕ;
f(2n + 1) = -n for every n ∈ ℕ.
This is clearly a one-to-one correspondence.  You seem to be saying that because I'm dealing with infinite sets, there must be some possibility that something about this correspondence will break down somewhere.  Infinite sets aren't really that special.
No need to be mean about it. I'll get back to this this afternoon though. I'm sure it's a problem with my delivery because you aren't understanding my issue at all. I may still be wrong but we aren't connecting.
Love is like a magic penny
 if you hold it tight you won't have any
if you give it away you'll have so many
they'll be rolling all over the floor