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Topic: infinities and the mind-dependence of maths (split from Newton etc.) (Read 963 times) previous topic - next topic - Topic derived from Newton, Copernicus, G...

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infinities and the mind-dependence of maths (split from Newton etc.)
NOTES ON MINE ...
1) Notice the 2 * 22 on both upper arms ... aaaah ... symmetry
You really have no idea what that word means, do you?
Symmetry between what and what?  :dunno:
Quote
2) Notice the two 63's entering and exiting the PC
I notice they make no particular sense.
PC * 63*20 = EC ? ? ?
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3) Notice the "12-ness" associated with the PR, PC and ER
From which you conclude.... :dunno: 
Something to do with with Ancient Knowledge ?
Quote
4) You can enter from PR or PC and derive PI very accurately - 5 decimal places 
Oh? How does that work?
Anything to do with any Ancient Knowledge ?
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5) Everything is whole numbers
unlike the real world. This is supposed to be a plus?
I wonder if Bluffy realizes there are as many non-whole numbers as there are whole numbers.
More.  Cantor.
If there are an infinity of numbers, then there will be an infinity of any portion of those numbers.
While it may seem like this is intuitively true, Cantor did manage to demonstrate that the rationals are countably infinite while the reals are uncountably infinite. Although, once you are talking about infinities, logic may go out the window. But that is part of the mathematical dogma of today.
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Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #1
I know I'm off-topic, but
I wonder if Bluffy realizes there are as many non-whole numbers as there are whole numbers.
this isn't true.

The integers are countable.  The rational numbers are also countable.  The irrational numbers are uncountable.

(I see JonF got to this point before I did.)
OK, so how many integers are there? Count them if you like. Or estimate the number.
Then tell me how many rational numbers there are.
How many integers are there:  ℶ0.
How many rational numbers are there:  also ℶ0.
How many real numbers are there:  ℶ1 (which is greater than ℶ0).

There are many different sizes of infinity.
But there are no sizes between the rationals and the reals. Right? I have never really gotten the nuances of mathematical descriptions of infinities. It all seems rather predicated on shaky premises. Although I do understand the rationale for the continuum hypothesis.

ETA: The symbol that is displayed in my browser looks nothing like an aleph.
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Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #2
I know I'm off-topic, but
I wonder if Bluffy realizes there are as many non-whole numbers as there are whole numbers.
this isn't true.

The integers are countable.  The rational numbers are also countable.  The irrational numbers are uncountable.

(I see JonF got to this point before I did.)
OK, so how many integers are there? Count them if you like. Or estimate the number.
Then tell me how many rational numbers there are.
How many integers are there:  ℶ0.
How many rational numbers are there:  also ℶ0.
How many real numbers are there:  ℶ1 (which is greater than ℶ0).

There are many different sizes of infinity.
But there are no sizes between the rationals and the reals. Right? I have never really gotten the nuances of mathematical descriptions of infinities. It all seems rather predicated on shaky premises. Although I do understand the rationale for the continuum hypothesis.

ETA: The symbol that is displayed in my browser looks nothing like an aleph.

Well, it's to do with whether you can map each members of one infinite set onto just one member of the other infinite set.  If you can, then the two infinities are the same size.  If you have spares in one set, then the set with spares is bigger.

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Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #3
I know I'm off-topic, but
I wonder if Bluffy realizes there are as many non-whole numbers as there are whole numbers.
this isn't true.

The integers are countable.  The rational numbers are also countable.  The irrational numbers are uncountable.

(I see JonF got to this point before I did.)
OK, so how many integers are there? Count them if you like. Or estimate the number.
Then tell me how many rational numbers there are.
How many integers are there:  ℶ0.
How many rational numbers are there:  also ℶ0.
How many real numbers are there:  ℶ1 (which is greater than ℶ0).

There are many different sizes of infinity.
But there are no sizes between the rationals and the reals. Right? I have never really gotten the nuances of mathematical descriptions of infinities. It all seems rather predicated on shaky premises. Although I do understand the rationale for the continuum hypothesis.

ETA: The symbol that is displayed in my browser looks nothing like an aleph.

Well, it's to do with whether you can map each members of one infinite set onto just one member of the other infinite set.  If you can, then the two infinities are the same size.  If you have spares in one set, then the set with spares is bigger.
right. That gives you two infinities: countable and uncountable. However, it also assumes that infinities behave the same way forever. I'm not sure that's a valid assumption. It's awfully hard to test at any rate.
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Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #4
How many integers are there:  ℶ0.
How many rational numbers are there:  also ℶ0.
How many real numbers are there:  ℶ1 (which is greater than ℶ0).

There are many different sizes of infinity.
But there are no sizes between the rationals and the reals. Right?
Thus saith the continuum hypothesis.  I read somewhere that depending on which foundational axioms you take for set theory, the continuum hypothesis can be either true or false, and it makes no difference to most of mathematics.  You can pretty much take it or leave it.
Quote from: Testy
I have never really gotten the nuances of mathematical descriptions of infinities. It all seems rather predicated on shaky premises.
Much of that stuff can be derived without any shaky (IMO) premises.  The basic idea is that two sets are considered to be of the same "size" if there exists a one-to-one correspondence between them.  That's how we compare finite sets, so why not extend the same idea to infinite sets?

And if you make an arbitrary one-to-one function from the set of natural numbers into the set of real numbers, one can always find a real number that is outside the range of that function.  So such a one-to-one correspondence does not exist.  And therefore the set of real numbers is bigger than the set of natural numbers.  Nothing shaky there AFAICS.
Quote from: Testy
Although I do understand the rationale for the continuum hypothesis.
Then you're ahead of me in that respect. :)
Quote from: Testy
ETA: The symbol that is displayed in my browser looks nothing like an aleph.
Yeah I was using beth-numbers rather than aleph-numbers.  They're easier to understand.

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Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #5
ETA: It also assumes that infinities are enclosable within sets.
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Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #6
How many integers are there:  ℶ0.
How many rational numbers are there:  also ℶ0.
How many real numbers are there:  ℶ1 (which is greater than ℶ0).

There are many different sizes of infinity.
But there are no sizes between the rationals and the reals. Right?
Thus saith the continuum hypothesis.  I read somewhere that depending on which foundational axioms you take for set theory, the continuum hypothesis can be either true or false, and it makes no difference to most of mathematics.  You can pretty much take it or leave it.
Quote from: Testy
I have never really gotten the nuances of mathematical descriptions of infinities. It all seems rather predicated on shaky premises.
Much of that stuff can be derived without any shaky (IMO) premises.  The basic idea is that two sets are considered to be of the same "size" if there exists a one-to-one correspondence between them.  That's how we compare finite sets, so why not extend the same idea to infinite sets?

And if you make an arbitrary one-to-one function from the set of natural numbers into the set of real numbers, one can always find a real number that is outside the range of that function.  So such a one-to-one correspondence does not exist.  And therefore the set of real numbers is bigger than the set of natural numbers.  Nothing shaky there AFAICS.
Quote from: Testy
Although I do understand the rationale for the continuum hypothesis.
Then you're ahead of me in that respect. :)
Quote from: Testy
ETA: The symbol that is displayed in my browser looks nothing like an aleph.
Yeah I was using beth-numbers rather than aleph-numbers.  They're easier to understand.
Right. I don't understand beth numbers. That was my point. Sorry about not being explicit.
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Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #7
right. That gives you two infinities: countable and uncountable. However, it also assumes that infinities behave the same way forever. I'm not sure that's a valid assumption. It's awfully hard to test at any rate.
Let X be a set.

Let ℙ(X) denote the power set of X (i.e., the set of all subsets of X).

Then it's easy to prove that ℙ(X) is bigger than X.

So ℕ is smaller than ℙ(ℕ), which is smaller than ℙ(ℙ(ℕ)), which is smaller than ℙ(ℙ(ℙ(ℕ))), and so on.

I'm using ℕ to denote the set of natural numbers, the quintessentially countably-infinite set.)

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Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #8
Right. I don't understand beth numbers. That was my point. Sorry about not being explicit.
See here for the beginning of a little series of posts saying what little I know about the cardinality of sets.

See here for what (very) little I know about aleph-numbers and beth-numbers.

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Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #9
right. That gives you two infinities: countable and uncountable. However, it also assumes that infinities behave the same way forever. I'm not sure that's a valid assumption. It's awfully hard to test at any rate.
Let X be a set.

Let ℙ(X) denote the power set of X (i.e., the set of all subsets of X).

Then it's easy to prove that ℙ(X) is bigger than X.

So ℕ is smaller than ℙ(ℕ), which is smaller than ℙ(ℙ(ℕ)), which is smaller than ℙ(ℙ(ℙ(ℕ))), and so on.

I'm using ℕ to denote the set of natural numbers, the quintessentially countably-infinite set.)
Ok, I want to try to explain my problem with this logic. I do understand set theory in a basic sense  (I took an excellent logic class long ago where it was covered in some detail at any rate) but I think it has a faulty premise which basically pervades mathematics in general. Power sets bear a more than superficial relationship to Russell's paradox so I will use a basic formulation of that since it's possible to use language rather than symbols I don't want to look up.

Take a set of all the rational numbers. That is an operation which encapsulates an infinity. Then you can say that the set cannot be a member of itself so you get into the neverending recursive loop. But before you took that set, it didn't exist. So when you took that set, you created something which is iterative. If you want to do it again, then you need to include the previous set but only if you want to do it again. The iterative nature of the problem is the problem. The idea of infinity assumes a god's eye view which simply doesn't exist anywhere, not even in maths.
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Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #10
Right. I don't understand beth numbers. That was my point. Sorry about not being explicit.
See here for the beginning of a little series of posts saying what little I know about the cardinality of sets.

See here for what (very) little I know about aleph-numbers and beth-numbers.

thanks
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Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #11
Take a set of all the rational numbers. That is an operation which encapsulates an infinity. Then you can say that the set cannot be a member of itself so you get into the neverending recursive loop. But before you took that set, it didn't exist. So when you took that set, you created something which is iterative. If you want to do it again, then you need to include the previous set but only if you want to do it again. The iterative nature of the problem is the problem. The idea of infinity assumes a god's eye view which simply doesn't exist anywhere, not even in maths.
I can't begin to make head or tail of any of this.

I don't know what you mean by "taking" a set.  If I tell someone to "take" a set, I'd just mean "consider" that set (because I'm going to be talking about it).  How is that an "operation"?

What does it mean to "encapsulate" an infinity?

What recursive loop are you talking about?

I don't see how the existence of a set can have anything to do with what I happen to be doing.  When I "take" a set (in my sense of the word), I'm just considering something that's there anyway, I'm not creating anything.

How does infinity assume a god's eye view?  I'm not a god, and I can think about infinite sets without any problem.

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Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #12
Take a set of all the rational numbers. That is an operation which encapsulates an infinity. Then you can say that the set cannot be a member of itself so you get into the neverending recursive loop. But before you took that set, it didn't exist. So when you took that set, you created something which is iterative. If you want to do it again, then you need to include the previous set but only if you want to do it again. The iterative nature of the problem is the problem. The idea of infinity assumes a god's eye view which simply doesn't exist anywhere, not even in maths.
I can't begin to make head or tail of any of this.

I don't know what you mean by "taking" a set.  If I tell someone to "take" a set, I'd just mean "consider" that set (because I'm going to be talking about it).  How is that an "operation"?

What does it mean to "encapsulate" an infinity?

What recursive loop are you talking about?

I don't see how the existence of a set can have anything to do with what I happen to be doing.  When I "take" a set (in my sense of the word), I'm just considering something that's there anyway, I'm not creating anything.

How does infinity assume a god's eye view?  I'm not a god, and I can think about infinite sets without any problem.
A set is something one needs to create. It is an operation that a mind performs on an abstraction. The operation of the mind is completely ignored in set theory and the objects are considered natural entities. Where in nature is a set? The answer is 'in a mind'. It is a mental operation.

EDIT: a set is an operation. The god's eye view is to assume all the iterations already exist but they don't and can't. The recursive loop is the paradox of taking a set of all sets. It can be done but not if one takes a god's eye view and includes all the sets that will be created. For each iteration though, that set is creatable.
  • Last Edit: September 07, 2016, 10:25:42 AM by Testy Calibrate
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Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #13
Thread is now about infinities.
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Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #14
A set is something one needs to create.
I disagree.  I'd suggest that it's a matter of perception, not creation.
Quote from: Testy
It is an operation that a mind performs on an abstraction.
Even in your account, you assume the prior existence of something (here you call it an "abstraction") on which the mind can perform this mysterious "operation".  Just take the set to be that "abstraction", and dispense with this superfluous notion of "creation".
Quote from: Testy
The operation of the mind is completely ignored in set theory and the objects are considered natural entities.
"Natural" strikes me as a bit of a red herring here.
Quote from: Testy
Where in nature is a set?
The question presumes that a set is the sort of thing that can have a "where".  I would dispute that presumption.

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Reply #15
EDIT: a set is an operation.
Since when?
Quote from: Testy
The god's eye view is to assume all the iterations already exist but they don't and can't.
Iterations of what?
Quote from: Testy
The recursive loop is the paradox of taking a set of all sets.
Paradoxes and recursive loops are not the same thing.  I'm having a lot of trouble following you here.

If we consider the set of all ZF-compatible sets, and thereby avoid Russell's Paradox, where is the problem?

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Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #16
A set is something one needs to create.
I disagree.  I'd suggest that it's a matter of perception, not creation.
Quote from: Testy
It is an operation that a mind performs on an abstraction.
Even in your account, you assume the prior existence of something (here you call it an "abstraction") on which the mind can perform this mysterious "operation".  Just take the set to be that "abstraction", and dispense with this superfluous notion of "creation".
but math is constructed. The abstraction of number must happen before the abstraction of an operation. And the construction of the idea of a set must follow from other constructed elements. I guess this is somewhere along the, uh, continuum, of the question of whether math is invented or discovered, but that isn't really my point. My point is that each of the constructs must happen sequentially in order to be valid. Once a turing machine solves a particular problem, it becomes one of the set that is solved. Until it does, that problem is unsolved. The fact that those imaginary machines are actually described as machines and that the operations are performed imaginatorily (made that word up and added it to spell check on the spot!) yet still described as operations speaks to that issue. Also, that there are truths about a system that are not provable from within the system follows the same logic. Once seen to be true, they are within the system. There is no god's eye view of all the possible truths, all the possible sets, all the possible divisions of the number line, none of those. They are all relating to our strangely finite machinations. Maybe you could even throw in the difference between infinite potential as in a random-ish pick from the reals and the then notable lack of infinities which are actually selectable as in however long you count or divide or whatever, you are going to have a finite number of operations and results, regardless how infinite the reals may or may not be. 
Quote
Quote from: Testy
The operation of the mind is completely ignored in set theory and the objects are considered natural entities.
"Natural" strikes me as a bit of a red herring here.
Well, I don't think it is. I think it's fairly central to the issue. Either a set is constructed or it already exists. I don't think there is a valid case that it already exists without invoking god since a brain has to have constructed it.

Quote
Quote from: Testy
Where in nature is a set?
The question presumes that a set is the sort of thing that can have a "where".  I would dispute that presumption.
Exactly.
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Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #17
It has always bothered me that the AoC has no mechanism too. Maybe that is related.
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Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #18
EDIT: a set is an operation.
Since when?
Quote from: Testy
The god's eye view is to assume all the iterations already exist but they don't and can't.
Iterations of what?
Quote from: Testy
The recursive loop is the paradox of taking a set of all sets.
Paradoxes and recursive loops are not the same thing.  I'm having a lot of trouble following you here.

If we consider the set of all ZF-compatible sets, and thereby avoid Russell's Paradox, where is the problem?

I used Russell's Paradox because it exemplifies the problem of taking a god's eye view of an iterative process.
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Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #19
but math is constructed.
In a sense.  But in that case we have to draw a distinction:  There's "math" the human activity, and then there are all the entities that are the object of that study (which we could collectively call "math").  Map versus territory.

I won't dispute that "math" in the former sense is constructed.  But it doesn't follow that any of the elements of "math" in the latter sense are themselves "constructed" (in the same sense of that word).
Quote from: Testy
And the construction of the idea of a set must follow from other constructed elements.
The construction of the idea of a set is different from the construction of a set.  The former is a mental operation (hence your word "idea"); it doesn't follow that the latter is.
Quote from: Testy
Either a set is constructed or it already exists. I don't think there is a valid case that it already exists without invoking god since a brain has to have constructed it.
But "since a brain has to have constructed it" is exactly the point of dispute.  Your argument looks glaringly circular to me.

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Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #20
I used Russell's Paradox because it exemplifies the problem of taking a god's eye view of an iterative process.
I don't follow you.

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Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #21
I'll never be a good materialist.  My instincts are too Platonic.  :(

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Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #22
I used Russell's Paradox because it exemplifies the problem of taking a god's eye view of an iterative process.
I don't follow you.
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.
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Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #23
I'll never be a good materialist.  My instincts are too Platonic.  :(
Funny, I am terrible at both.
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Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #24
but math is constructed.
In a sense.  But in that case we have to draw a distinction:  There's "math" the human activity, and then there are all the entities that are the object of that study (which we could collectively call "math").  Map versus territory.

I won't dispute that "math" in the former sense is constructed.  But it doesn't follow that any of the elements of "math" in the latter sense are themselves "constructed" (in the same sense of that word).
Quote from: Testy
And the construction of the idea of a set must follow from other constructed elements.
The construction of the idea of a set is different from the construction of a set.  The former is a mental operation (hence your word "idea"); it doesn't follow that the latter is.
Quote from: Testy
Either a set is constructed or it already exists. I don't think there is a valid case that it already exists without invoking god since a brain has to have constructed it.
But "since a brain has to have constructed it" is exactly the point of dispute.  Your argument looks glaringly circular to me.
Hmm. This isn't obvious to me. The idea of a set is an abstraction. A set of {x} is an operation. No?

ETA: or are you saying this isn't a case where an excluded middle applies?
  • Last Edit: September 07, 2016, 11:47:26 AM by Testy Calibrate
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