0d Lines and Circles

What is the basis for reason? And mathematics?

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wtf
Posts: 913
Joined: Tue Sep 08, 2015 11:36 pm

Re: 0d Lines and Circles

Post by wtf » Thu Oct 17, 2019 7:45 pm

Atla wrote:
Thu Oct 17, 2019 6:24 pm
Actually, nothing in actual science rests on this fallacy, in case you haven't figured that out either.
I agree that the universe keeps on existing and the galaxies keep flying apart and bowling balls keep falling to earth without intervention from people. No theory of physics is required for the universe to exist. The theories come after, and are only approximate and always historically contingent.

It's also true that our historically contingent theories of the physical world do rest on conventional math, which is based on conventional set theory. So the burden would be on you to redevelop all of math and physical science based on your alternate ideas. Or else adopt the position that, "The universe is what it is and we don't need no stinkin' theory." That's cool too, I like it here in my cave. Mastodon for dinner, yum.

Just so I can understand your objection, can I ask some questions? Let's start with basic set theory. Do you agree or disagree that as sets, {a, b, c} = {a} ∪ {b} ∪ {c}?

How about infinite sets? Do you agree that the set of natural numbers {0, 1, 2, 3, ...} is equal to the union of the singleton set {n} over all natural number n?

I'm just trying to understand the depth of your objection to the union operation. I'd like to better understand why you don't think a set is the union of the singletons of its elements.

Atla
Posts: 2569
Joined: Fri Dec 15, 2017 8:27 am

Re: 0d Lines and Circles

Post by Atla » Thu Oct 17, 2019 7:52 pm

wtf wrote:
Thu Oct 17, 2019 7:45 pm
Atla wrote:
Thu Oct 17, 2019 6:24 pm
Actually, nothing in actual science rests on this fallacy, in case you haven't figured that out either.
I agree that the universe keeps on existing and the galaxies keep flying apart and bowling balls keep falling to earth without intervention from people. No theory of physics is required for the universe to exist. The theories come after, and are only approximate and always historically contingent.

It's also true that our historically contingent theories of the physical world do rest on conventional math, which is based on conventional set theory. So the burden would be on you to redevelop all of math and physical science based on your alternate ideas. Or else adopt the position that, "The universe is what it is and we don't need no stinkin' theory." That's cool too, I like it here in my cave. Mastodon for dinner, yum.

Just so I can understand your objection, can I ask some questions? Let's start with basic set theory. Do you agree or disagree that as sets, {a, b, c} = {a} ∪ {b} ∪ {c}?

How about infinite sets? Do you agree that the set of natural numbers {0, 1, 2, 3, ...} is equal to the union of the singleton set {n} over all natural number n?

I'm just trying to understand the depth of your objection to the union operation. I'd like to better understand why you don't think a set is the union of the singletons of its elements.
I didn't object to the union operation. To be honest I don't even see what sets have to do with the question of 0d points vs 1d lines.

Eodnhoj7
Posts: 5670
Joined: Mon Mar 13, 2017 3:18 am

Re: 0d Lines and Circles

Post by Eodnhoj7 » Thu Oct 17, 2019 8:02 pm

Atla wrote:
Thu Oct 17, 2019 6:38 pm
Eodnhoj7 wrote:
Thu Oct 17, 2019 6:33 pm
So a unit such as a meter is not composed of centimeters or millimeters?
A unit itself is just a concept and isn't 'actually' made of anything, whether you divide it into smaller parts or not. :roll:
Predictable cog.

You didn't answer the second question; if there are no units in the real world, then there are no real measurements then either?


Please, wise one, enlighten us since you know so much.

Atla
Posts: 2569
Joined: Fri Dec 15, 2017 8:27 am

Re: 0d Lines and Circles

Post by Atla » Thu Oct 17, 2019 8:05 pm

Eodnhoj7 wrote:
Thu Oct 17, 2019 8:02 pm
Atla wrote:
Thu Oct 17, 2019 6:38 pm
Eodnhoj7 wrote:
Thu Oct 17, 2019 6:33 pm
So a unit such as a meter is not composed of centimeters or millimeters?
A unit itself is just a concept and isn't 'actually' made of anything, whether you divide it into smaller parts or not. :roll:
Predictable cog.

You didn't answer the second question; if there are no units in the real world, then there are no real measurements then either?


Please, wise one, enlighten us since you know so much.
I skipped it on purpose because it was a too stupid question, and now you very predictably think that you got me.

Idiot.

It's like when the theist asks the atheist: "but how do you pretend God away?"

wtf
Posts: 913
Joined: Tue Sep 08, 2015 11:36 pm

Re: 0d Lines and Circles

Post by wtf » Thu Oct 17, 2019 8:23 pm

Atla wrote:
Thu Oct 17, 2019 7:52 pm
I didn't object to the union operation. To be honest I don't even see what sets have to do with the question of 0d points vs 1d lines.
Oh ok. Good point. Perhaps I didn't make the connection clear.

In modern math -- again, this is a historically contingent theory made up by humans, and has arguably nothing (at least for sake of this conversation) to do with the actual universe -- but in modern math, the real numbers form a set. A set is informally thought of as a collection of objects. So for example the set of natural numbers {0, 1, 2, 3, ...} is a collection containing each of the numbers 0, 1, 2, 3, ...

Because of the way the union operation is defined we can write

{0, 1, 2, ...} = {0} ∪ {1} ∪ {2} ∪ ...

where the little '∪' is the union operation.

In face any set whatsoever is equal to the union of its singleton subsets. So the unit interval, the set of real numbers between 0 and 1, is the union of the singletons of its elements. This is incontrovertible unless you are prepared to throw out the whole of set theory on which modern math is based. [Again you are free to do so, but you must then rebuild math and physics etc. according to your new theory. The burden is on you].

So the unit interval, which has length 1, is equal to the union of infinitely many singleton sets {x} where x is some real number.

@surreptitious57 claimed that therefore the unit interval must have length 0. This of course contradicts high school math. One can hardly build science on a foundation that claims that the distance from point 0 to the point 1 is zero.

What exactly is your view on how to handle the fact that the unit interval has length 1, and it's the union of all these singleton sets each containing a point of length 0?

There is by the way a mathematical way of handling this. We don't require lengths to add up for arbitrary infinite sets; only for countably infinite sets. In probability and measure theory this is known as countable additivity.

https://en.wikipedia.org/wiki/Sigma_additivity

The reason we reject arbitrary additivity is exactly because of this paradox: that the unit interval has length 1, and each individual point has length 0.

I do in fact regard this as a mystery. Math doesn't explain it; math only finesses the issue by only requiring countable additivity for lengths and probabilities.

We do have a formalism in calculus that the integral from 0 to 1 of dx is 1. That expresses the fact that informally "adding up all the 0-length points gives you 1." We have a formalism but not truly a philosophical explanation, in my opinion.

What's your take?

Eodnhoj7
Posts: 5670
Joined: Mon Mar 13, 2017 3:18 am

Re: 0d Lines and Circles

Post by Eodnhoj7 » Thu Oct 17, 2019 8:26 pm

Atla wrote:
Thu Oct 17, 2019 8:05 pm
Eodnhoj7 wrote:
Thu Oct 17, 2019 8:02 pm
Atla wrote:
Thu Oct 17, 2019 6:38 pm

A unit itself is just a concept and isn't 'actually' made of anything, whether you divide it into smaller parts or not. :roll:
Predictable cog.

You didn't answer the second question; if there are no units in the real world, then there are no real measurements then either?


Please, wise one, enlighten us since you know so much.
I skipped it on purpose because it was a too stupid question, and now you very predictably think that you got me.

Idiot.

It's like when the theist asks the atheist: "but how do you pretend God away?"
Again predictable, with the fear.

So if no units are real, then all measurements are not real, and your argument as a measurement is not real either.

Eodnhoj7
Posts: 5670
Joined: Mon Mar 13, 2017 3:18 am

Re: 0d Lines and Circles

Post by Eodnhoj7 » Thu Oct 17, 2019 8:27 pm

wtf wrote:
Thu Oct 17, 2019 8:23 pm
Atla wrote:
Thu Oct 17, 2019 7:52 pm
I didn't object to the union operation. To be honest I don't even see what sets have to do with the question of 0d points vs 1d lines.
Oh ok. Good point. Perhaps I didn't make the connection clear.

In modern math -- again, this is a historically contingent theory made up by humans, and has arguably nothing (at least for sake of this conversation) to do with the actual universe -- but in modern math, the real numbers form a set. A set is informally thought of as a collection of objects. So for example the set of natural numbers {0, 1, 2, 3, ...} is a collection containing each of the numbers 0, 1, 2, 3, ...

Because of the way the union operation is defined we can write

{0, 1, 2, ...} = {0} ∪ {1} ∪ {2} ∪ ...

where the little '∪' is the union operation.

In face any set whatsoever is equal to the union of its singleton subsets. So the unit interval, the set of real numbers between 0 and 1, is the union of the singletons of its elements. This is incontrovertible unless you are prepared to throw out the whole of set theory on which modern math is based. [Again you are free to do so, but you must then rebuild math and physics etc. according to your new theory. The burden is on you].

So the unit interval, which has length 1, is equal to the union of infinitely many singleton sets {x} where x is some real number.

@surreptitious57 claimed that therefore the unit interval must have length 0. This of course contradicts high school math. One can hardly build science on a foundation that claims that the distance from point 0 to the point 1 is zero.

What exactly is your view on how to handle the fact that the unit interval has length 1, and it's the union of all these singleton sets each containing a point of length 0?

There is by the way a mathematical way of handling this. We don't require lengths to add up for arbitrary infinite sets; only for countably infinite sets. In probability and measure theory this is known as countable additivity.

https://en.wikipedia.org/wiki/Sigma_additivity

The reason we reject arbitrary additivity is exactly because of this paradox: that the unit interval has length 1, and each individual point has length 0.

I do in fact regard this as a mystery. Math doesn't explain it; math only finesses the issue by only requiring countable additivity for lengths and probabilities.

We do have a formalism in calculus that the integral from 0 to 1 of dx is 1. That expresses the fact that informally "adding up all the 0-length points gives you 1." We have a formalism but not truly a philosophical explanation, in my opinion.

What's your take?
Tell me...if two or more points occur....is there a length between them?

Atla
Posts: 2569
Joined: Fri Dec 15, 2017 8:27 am

Re: 0d Lines and Circles

Post by Atla » Thu Oct 17, 2019 8:32 pm

Eodnhoj7 wrote:
Thu Oct 17, 2019 8:26 pm
Atla wrote:
Thu Oct 17, 2019 8:05 pm
Eodnhoj7 wrote:
Thu Oct 17, 2019 8:02 pm


Predictable cog.

You didn't answer the second question; if there are no units in the real world, then there are no real measurements then either?


Please, wise one, enlighten us since you know so much.
I skipped it on purpose because it was a too stupid question, and now you very predictably think that you got me.

Idiot.

It's like when the theist asks the atheist: "but how do you pretend God away?"
Again predictable, with the fear.

So if no units are real, then all measurements are not real, and your argument as a measurement is not real either.
An argument is not a measurement, and the whole thing has nothing to do with realness. And don't project your fear on me just because we are nearing the point where your schizophrenic world collapses. :)

Eodnhoj7
Posts: 5670
Joined: Mon Mar 13, 2017 3:18 am

Re: 0d Lines and Circles

Post by Eodnhoj7 » Thu Oct 17, 2019 8:33 pm

Atla wrote:
Thu Oct 17, 2019 8:32 pm
Eodnhoj7 wrote:
Thu Oct 17, 2019 8:26 pm
Atla wrote:
Thu Oct 17, 2019 8:05 pm

I skipped it on purpose because it was a too stupid question, and now you very predictably think that you got me.

Idiot.

It's like when the theist asks the atheist: "but how do you pretend God away?"
Again predictable, with the fear.

So if no units are real, then all measurements are not real, and your argument as a measurement is not real either.
An argument is not a measurement, and the whole thing has nothing to do with realness. And don't project your fear on me just because we are nearing the point where your schizophrenic world collapses. :)
So measurements are not real?

Atla
Posts: 2569
Joined: Fri Dec 15, 2017 8:27 am

Re: 0d Lines and Circles

Post by Atla » Thu Oct 17, 2019 8:38 pm

wtf wrote:
Thu Oct 17, 2019 8:23 pm
Atla wrote:
Thu Oct 17, 2019 7:52 pm
I didn't object to the union operation. To be honest I don't even see what sets have to do with the question of 0d points vs 1d lines.
Oh ok. Good point. Perhaps I didn't make the connection clear.

In modern math -- again, this is a historically contingent theory made up by humans, and has arguably nothing (at least for sake of this conversation) to do with the actual universe -- but in modern math, the real numbers form a set. A set is informally thought of as a collection of objects. So for example the set of natural numbers {0, 1, 2, 3, ...} is a collection containing each of the numbers 0, 1, 2, 3, ...

Because of the way the union operation is defined we can write

{0, 1, 2, ...} = {0} ∪ {1} ∪ {2} ∪ ...

where the little '∪' is the union operation.

In face any set whatsoever is equal to the union of its singleton subsets. So the unit interval, the set of real numbers between 0 and 1, is the union of the singletons of its elements. This is incontrovertible unless you are prepared to throw out the whole of set theory on which modern math is based. [Again you are free to do so, but you must then rebuild math and physics etc. according to your new theory. The burden is on you].

So the unit interval, which has length 1, is equal to the union of infinitely many singleton sets {x} where x is some real number.

@surreptitious57 claimed that therefore the unit interval must have length 0. This of course contradicts high school math. One can hardly build science on a foundation that claims that the distance from point 0 to the point 1 is zero.

What exactly is your view on how to handle the fact that the unit interval has length 1, and it's the union of all these singleton sets each containing a point of length 0?

There is by the way a mathematical way of handling this. We don't require lengths to add up for arbitrary infinite sets; only for countably infinite sets. In probability and measure theory this is known as countable additivity.

https://en.wikipedia.org/wiki/Sigma_additivity

The reason we reject arbitrary additivity is exactly because of this paradox: that the unit interval has length 1, and each individual point has length 0.

I do in fact regard this as a mystery. Math doesn't explain it; math only finesses the issue by only requiring countable additivity for lengths and probabilities.

We do have a formalism in calculus that the integral from 0 to 1 of dx is 1. That expresses the fact that informally "adding up all the 0-length points gives you 1." We have a formalism but not truly a philosophical explanation, in my opinion.

What's your take?
I don't even know where to start..

You aren't really talking about 0d points and 1d lines.

The interval 0,1 is an interval on the numberline, it has no length.

Numbers on the number line are numbers, coordinates; not points.

There are infinitely many numbers between any two numbers. And if you start from 0 and keep adding 0, after infinitely many additions you are still at 0.

It also makes no sense to say that the 0,1 interval is the union of all subsets. We forgot that there we made a step before. First we took the interval, divided into infinitely many values. And then took the union of those values and claim that that's how we got the interval.

All in all I don't see a mistery here. We can't get 1d concepts from adding 0d concepts, period. A 0d point doesn't even have a value on the 1d numberline, because it's 0d. A 0d point isn't anywhere.

I only see a desperate attempt to connect everything with everything else in math, even when dimensionality prevents that.

wtf
Posts: 913
Joined: Tue Sep 08, 2015 11:36 pm

Re: 0d Lines and Circles

Post by wtf » Thu Oct 17, 2019 8:40 pm

Eodnhoj7 wrote:
Thu Oct 17, 2019 8:27 pm
Tell me...if two or more points occur....is there a length between them?
Yes, this is the fundamental idea of a metric space. Given any two points there's a function, called the metric, that gives the distance (or length) between them.

In the real numbers the distance between the points x and y is |x - y|. So the distance between -2 and 3 is 5. I hope you picked this up in school.

In the Euclidean plane, the distance between points is the Pythagorean distance, the square root of the sum of the squares of the differences between the respective coordinates of the points. So the distance between (0,0) and (1,1) is sqrt(2). Again this is taught to 16 year olds all over the world.

There are other notions of distance. For example suppose we are in a city with a grid of streets and avenues perpendicular to each other. Then the distance from first street and first avenue to second street and second avenue is two blocks. Because you have to drive horizontally one unit and then vertically one unit.

This is called the taxicab metric or the Manhattan metric.

So yes, there's an entire mathematical subject of distances, and lots of different ways of assigning distances.

https://en.wikipedia.org/wiki/Metric_space

https://en.wikipedia.org/wiki/Taxicab_geometry

Atla
Posts: 2569
Joined: Fri Dec 15, 2017 8:27 am

Re: 0d Lines and Circles

Post by Atla » Thu Oct 17, 2019 8:42 pm

Eodnhoj7 wrote:
Thu Oct 17, 2019 8:33 pm
Atla wrote:
Thu Oct 17, 2019 8:32 pm
Eodnhoj7 wrote:
Thu Oct 17, 2019 8:26 pm

Again predictable, with the fear.

So if no units are real, then all measurements are not real, and your argument as a measurement is not real either.
An argument is not a measurement, and the whole thing has nothing to do with realness. And don't project your fear on me just because we are nearing the point where your schizophrenic world collapses. :)
So measurements are not real?
You keep saying that we "observe" things, but observation is just a metaphor, it has no special role.

If you reify "measurement" into something that somehow drives the world, then no, in that sense, measurements are not real.

Eodnhoj7
Posts: 5670
Joined: Mon Mar 13, 2017 3:18 am

Re: 0d Lines and Circles

Post by Eodnhoj7 » Thu Oct 17, 2019 8:46 pm

wtf wrote:
Thu Oct 17, 2019 8:40 pm
Eodnhoj7 wrote:
Thu Oct 17, 2019 8:27 pm
Tell me...if two or more points occur....is there a length between them?
Yes, this is the fundamental idea of a metric space. Given any two points there's a function, called the metric, that gives the distance (or length) between them.

In the real numbers the distance between the points x and y is |x - y|. So the distance between -2 and 3 is 5. I hope you picked this up in school.

In the Euclidean plane, the distance between points is the Pythagorean distance, the square root of the sum of the squares of the differences between the respective coordinates of the points. So the distance between (0,0) and (1,1) is sqrt(2). Again this is taught to 16 year olds all over the world.

There are other notions of distance. For example suppose we are in a city with a grid of streets and avenues perpendicular to each other. Then the distance from first street and first avenue to second street and second avenue is two blocks. Because you have to drive horizontally one unit and then vertically one unit.

This is called the taxicab metric or the Manhattan metric.

So yes, there's an entire mathematical subject of distances, and lots of different ways of assigning distances.

https://en.wikipedia.org/wiki/Metric_space

https://en.wikipedia.org/wiki/Taxicab_geometry
So if a line is length between two points, and a line is composed of infinite points, a line is composed of infinite lengths?

Eodnhoj7
Posts: 5670
Joined: Mon Mar 13, 2017 3:18 am

Re: 0d Lines and Circles

Post by Eodnhoj7 » Thu Oct 17, 2019 8:48 pm

Atla wrote:
Thu Oct 17, 2019 8:42 pm
Eodnhoj7 wrote:
Thu Oct 17, 2019 8:33 pm
Atla wrote:
Thu Oct 17, 2019 8:32 pm

An argument is not a measurement, and the whole thing has nothing to do with realness. And don't project your fear on me just because we are nearing the point where your schizophrenic world collapses. :)
So measurements are not real?
You keep saying that we "observe" things, but observation is just a metaphor, it has no special role.

If you reify "measurement" into something that somehow drives the world, then no, in that sense, measurements are not real.
A metaphor for what exactly?

So if I measure a piece of stone, and further measure these multiple stones to build a house, the measurement did not change one state of the phenomena (stone) into another (house)?

Atla
Posts: 2569
Joined: Fri Dec 15, 2017 8:27 am

Re: 0d Lines and Circles

Post by Atla » Thu Oct 17, 2019 8:53 pm

Eodnhoj7 wrote:
Thu Oct 17, 2019 8:48 pm
Atla wrote:
Thu Oct 17, 2019 8:42 pm
Eodnhoj7 wrote:
Thu Oct 17, 2019 8:33 pm

So measurements are not real?
You keep saying that we "observe" things, but observation is just a metaphor, it has no special role.

If you reify "measurement" into something that somehow drives the world, then no, in that sense, measurements are not real.
A metaphor for what exactly?

So if I measure a piece of stone, and further measure these multiple stones to build a house, the measurement did not change one state of the phenomena (stone) into another (house)?
A metaphor for pretending something into the real world that's not actually there.

For example you pretend that the phenomena have states like stone and house, even though there are no distinct phenomena and the universe has been merely rearranged, so to speak.

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