Heaven and hell are not just "illusory"

[godelian]

I didn't attack mathemathics

If you reject the provable existence of nonstandard numbers as a delusion then you reject arithmetic theory.

just a philosophical interpretation of mathemathics.

Nonstandard numbers are not "just a philosophical interpretation". They are provable from the compactness theorem, from the incompleteness theorems, and from ultraproducts. As Gödel pointed out, it requires mathematical intuition and special ability to perceive nonstandard numbers directly and to fully understand why they are there.

Victoria Gitman has an interesting lecture in which she describes the shape of the arithmetical multiverse:

https://victoriagitman.github.io/talks/ ... metic.html

Order-wise, these models look like the natural numbers followed by densely many copies of the integers: N followed by Q-many copies of Z (see the slides for explanation).

She also points out that there exist indiscernible numbers:

In particular, a nonstandard model of arithmetic can have indiscernible numbers that share all the same properties. Indeed, the automorphism groups of nonstandard models have some remarkable properties themselves (for details, see [2]).

This is actually nothing unusual. Through Richard's paradox (1905) undefinable numbers had already been discovered.

There are also ineffable numbers: https://en.wikipedia.org/wiki/Ineffable_cardinal

Because all of this is provable, it is mathematics and not mere "philosophical interpretation".

[Atla]

I didn't attack mathemathics

If you reject the provable existence of nonstandard numbers as a delusion then you reject arithmetic theory.

just a philosophical interpretation of mathemathics.

Nonstandard numbers are not "just a philosophical interpretation". They are provable from the compactness theorem, from the incompleteness theorems, and from ultraproducts. As Gödel pointed out, it requires mathematical intuition and special ability to perceive nonstandard numbers directly and to fully understand why they are there.

Victoria Gitman has an interesting lecture in which she describes the shape of the arithmetical multiverse:

https://victoriagitman.github.io/talks/ ... metic.html

Order-wise, these models look like the natural numbers followed by densely many copies of the integers: N followed by Q-many copies of Z (see the slides for explanation).

She also points out that there exist indiscernible numbers:

In particular, a nonstandard model of arithmetic can have indiscernible numbers that share all the same properties. Indeed, the automorphism groups of nonstandard models have some remarkable properties themselves (for details, see [2]).

This is actually nothing unusual. Through Richard's paradox (1905) undefinable numbers had already been discovered.

There are also ineffable numbers: https://en.wikipedia.org/wiki/Ineffable_cardinal

Because all of this is provable, it is mathematics and not mere "philosophical interpretation".

You are fundamentally confused my friend. All numbers are abstractions, not just nonstandard numbers.

[godelian]

You are fundamentally confused my friend. All numbers are abstractions, not just nonstandard numbers.

These abstractions truly exist. They are not fictitious or illusory. That is where you are fundamentally confused.

When we explore these abstract Platonic worlds, everybody sees exactly the same things.
Well, you could also see nothing at all, if you lack the somewhat special ability to observe them.

[Atla]

You are fundamentally confused my friend. All numbers are abstractions, not just nonstandard numbers.

These abstractions truly exist. They are not fictitious or illusory. That is where you are fundamentally confused.

When we explore these abstract Platonic worlds, everybody sees exactly the same things.
Well, you could also see nothing at all, if you lack the somewhat special ability to observe them.

Again you are fundamentally confused. When we explore abstract Platonic "worlds", we quite probably see the same things because we live in the same world with the same behaviour. Where you are, nature behaves the same way where I am, and so on. So even our abstractions reflect the same things.

But then the psychologically weak / the delusional, get intoxicated by these abstract "worlds" and go on to believe that they have a concrete existence.

[godelian]

Again you are fundamentally confused. When we explore abstract Platonic "worlds", we quite probably see the same things because we live in the same world with the same behaviour. Where you are, nature behaves the same way where I am, and so on. So even our abstractions reflect the same things.

These Platonic abstractions are not the physical universe.

For example, the standard natural numbers can somewhat be mapped to the physical universe, but nonstandard numbers cannot. If you map the standard natural numbers to the physical universe, then you must map the nonstandard numbers to nonstandard physical universes.

But then the psychologically weak / the delusional, get intoxicated by these abstract "worlds" and go on to believe that they have a concrete existence.

Untalented people believe that intelligent people are crazy. If you see someone manipulating strange symbols using complex expressions, which you do not understand, not even to save yourself from drowning, then you will think that this person is "not normal". Of course, this person technically isn't. Normality is the same as mediocrity. There is something that makes this kind of individuals different, i.e. some special ability, as Gödel pointed out.

There is nothing wrong with the club. You are just not a member of it. In some way, we feel sorry for you. If you don't have it, then you will just have to accept that. Try another club. Maybe that works better for you.

[Atla]

Again you are fundamentally confused. When we explore abstract Platonic "worlds", we quite probably see the same things because we live in the same world with the same behaviour. Where you are, nature behaves the same way where I am, and so on. So even our abstractions reflect the same things.

These Platonic abstractions are not the physical universe.

For example, the standard natural numbers can somewhat be mapped to the physical universe, but nonstandard numbers cannot. If you map the standard natural numbers to the physical universe, then you must map the nonstandard numbers to nonstandard physical universes.

But then the psychologically weak / the delusional, get intoxicated by these abstract "worlds" and go on to believe that they have a concrete existence.

Untalented people believe that intelligent people are crazy. If you see someone manipulating strange symbols using complex expressions, which you do not understand, not even to save yourself from drowning, then you will think that this person is "not normal". Of course, this person technically isn't. Normality is the same as mediocrity. There is something that makes this kind of individuals different, i.e. some special ability, as Gödel pointed out.

There is nothing wrong with the club. You are just not a member of it. In some way, we feel sorry for you. If you don't have it, then you will just have to accept that. Try another club. Maybe that works better for you.

You can't be sure that standard numbers can be somehow mapped to our natural world and non-standard numbers can't be. You're just making shit up again in order to pretend to belong to some sort of intellectual club, even though this club is just about hallucinating apparently.

(and plot twist: I'm far more into multiverse philosophy than you are)

[godelian]

You can't be sure that standard numbers can be somehow mapped to our natural world and non-standard numbers can't be.

Natural numbers (N) can be mapped onto a local physical time line at the origin of the universe. Nonstandard numbers however, which are copies of Z, i.e. the integers, expand in both a positive and negative direction. It would suggest that physical time expands by progression but also simultaneously by regression.

In any case, standard and nonstandard numbers cannot be mapped simultaneously onto one physical universe because that would suggest physical expansion simultaneously in an infinite number of directions. Time (noticeably) expands. Timespace (theoretically) expands. You can fit in four directions but not an infinite number of directions.

Another problem is that nonstandard numbers do not start at zero but at an infinite ordinal. If you map them onto the physical universe, you end up with actual infinities. That wouldn't work particularly well. Furthermore, you cannot do arithmetic with nonstandard numbers. Time calculations would be impossible if mapped onto the physical universe:

Tennenbaum's theorem shows that for any countable non-standard model of Peano arithmetic there is no way to code the elements of the model as (standard) natural numbers such that either the addition or multiplication operation of the model is computable on the codes. This result was first obtained by Stanley Tennenbaum in 1959.

In this regard, Victoria Gitman writes:

So finally we have that the natural numbers are unique in one sense, it is the only model of the Peano axioms in which we can compute.

So, nonstandard numbers would simply be unusable in our physical universe.

You're just making shit up

Mapping the natural numbers onto the physical universe is what everybody regularly does. Even animals do that routinely. Of course, not everybody tries to map the entire collection of standard and nonstandard models.

However, the nonstandard models exist as much as the standard model.

Victoria Gitman proposes a projection that looks like the image in the link:

https://victoriagitman.github.io/talks/ ... metic.html

Order-wise, these models look like the natural numbers followed by densely many copies of the integers: N followed by Q-many copies of Z (see the slides for explanation).

Her representation cannot respect scale because every nonstandard model starts at another infinite ordinal. So, the gaps in between the models are infinitely large. She tries to represent the continuum hypothesis by leaving blank space in between the models. ( https://en.wikipedia.org/wiki/Continuum_hypothesis )

The complete explanation for the representation starts on slide 29 and finishes on slide 40.

So, no, neither Victoria Gitman nor myself are "making shit up". We both see exactly the same abstract Platonic world(s) in front of us. You cannot see it because you do not have the ability required. That is why you believe that it does not exist.

[Veritas Aequitas]

These Platonic abstractions are not the physical universe.

For example, the standard natural numbers can somewhat be mapped to the physical universe, but nonstandard numbers cannot. If you map the standard natural numbers to the physical universe, then you must map the nonstandard numbers to nonstandard physical universes.

You are really lost philosophically.

You keep harping on platonic realism when it has loads of criticisms.
https://en.wikipedia.org/wiki/Theory_of ... onic_Forms

What is critical is whether things are real or not.
If it is not real, then it is imaginary, fictitious and illusory.
The gold standard for reality is from the scientific framework and system [FS], say benchmarked at 100/100.
The platonic FS is rated at 10/100 [best estimate], thus cannot be realistic, so it is imaginary, fictitious and illusory, BUT it is nevertheless a useful illusion to facilitate in understanding the real.

You mentioned Kant somewhere.
Kant did not have any problems with numbers whether natural or non-natural like fractions as long as whatever is related to the numbers can be confirmed by experience and science scientifically as real, e.g. half-an-apple and the likes.

[Atla]

Natural numbers (N) can be mapped onto a local physical time line at the origin of the universe. Nonstandard numbers however, which are copies of Z, i.e. the integers, expand in both a positive and negative direction. It would suggest that physical time expands by progression but also simultaneously by regression.

In any case, standard and nonstandard numbers cannot be mapped simultaneously onto one physical universe because that would suggest physical expansion simultaneously in an infinite number of directions. Time (noticeably) expands. Timespace (theoretically) expands. You can fit in four directions but not an infinite number.

Circular/religious fantasy. Assuming that the universe had an origin "0" timeline point.

There's no reason to think that time "expands". How could relative time expand anyway?
There's only reason to think that humans can only exist in a direction of increasing entropy, which direction simpler minds like you might perceive as expanding time.

Another problem is that nonstandard numbers do not start at zero but at an infinite ordinal. If you map them onto the physical universe, you end up with actual infinities. That wouldn't work particularly well. Furthermore, you cannot do arithmetic with nonstandard numbers. Time calculations would be impossible if mapped onto the physical universe:

Tennenbaum's theorem shows that for any countable non-standard model of Peano arithmetic there is no way to code the elements of the model as (standard) natural numbers such that either the addition or multiplication operation of the model is computable on the codes. This result was first obtained by Stanley Tennenbaum in 1959.

In this regard, Victoria Gitman writes:

So finally we have that the natural numbers are unique in one sense, it is the only model of the Peano axioms in which we can compute.

So, nonstandard numbers would simply be unusable in our physical universe.

On the contrary, it could be that the physical universe is (fractal?) infinites upon infinites in every "direction" and "scale", and any finite computation is just a crude approximation, a reflection of the infinites upon infinites. Our finites are.. proportions of the infinites we are embedded in.

Mapping the natural numbers onto the physical universe is what everybody regularly does. Even animals do that routinely. Of course, not everybody tries to map the entire collection of standard and nonstandard models.

However, the nonstandard models exist as much as the standard model.

Victoria Gitman proposes a projection that looks like the image in the link:

Order-wise, these models look like the natural numbers followed by densely many copies of the integers: N followed by Q-many copies of Z (see the slides for explanation).

Her representation cannot respect scale because every nonstandard model starts at another infinite ordinal. So, the gaps in between the models are infinitely large. She tries to represent the continuum hypothesis by leaving enough blank space in between the models. ( https://en.wikipedia.org/wiki/Continuum_hypothesis )

The complete explanation for the representation starts on slide 29 and finishes on slide 40.

So, no, neither Victoria Gitman nor myself are "making shit up". We both see exactly the same abstract Platonic world(s) in front of us. You cannot see it because you do not have the ability required. That is why you believe that it does not exist.

Natural numbers come from repeating the number 1, and the number 1 is an expression of the abstract unit, which is an abstract concept of finitude. Of course everyone maps natural numbers onto the physical universe, and of course that doesn't mean that abstract worlds exist.

Look, I'm much more into multiverse philosophy than you are (but physical multiverse), I'm just saying that we can't prove it, and that abstract worlds don't exist.

[godelian]

Natural numbers come from repeating the number 1

Except for number zero.
The number zero is axiomatized to be a natural number in Peano arithmetic theory (PA).
In fact, construction of the system does not mention any other number by name.
Only zero has that privilege.

and the number 1 is an expression of the abstract unit, which is an abstract concept of finitude.

The number one is not mentioned explicitly in PA. Instead, PA axiomatizes the successor function S. Hence, number one is not considered interesting enough to mention in the construction plans.

For me, the natural numbers are strictly the standard model that interprets Peano arithmetic theory. In this precise context, I am able to look up things in a database that contains over a century worth of mathematical work. I consider any other definition to be poorly specified.

Look, I'm much more into multiverse philosophy than you are (but physical multiverse)

I am mostly interested in the arithmetical multiverse because it is a provable object.

Of course, I sometimes like to map it onto the physical universe because that is what everybody does with numbers. I have never said that such mapping would be provable. It is obvious that no mapping between the numbers and the physical universe can ever be provable.

Pythagoras believed in sacred mathematics and thought that the universe could be understood through numbers.

Of course, I do not believe in "sacred" mathematics.

However, I suspect that there are important structural similarities between the arithmetical multiverse and the one of the physical universe.

I'm just saying that we can't prove it

What exactly is unprovable?

The arithmetical multiverse is perfectly provable.

The mapping between the arithmetical multiverse and the one of the physical universe, on the other hand, cannot be provable, if only because there is absolutely nothing provable about the physical universe.

abstract worlds don't exist.

Every member of the club sees exactly the same things.
How is that possible if all of us just invented them?
How do we manage to coordinate a common fictitious illusion so precisely?
So, for people who do not have the ability to see them, they do indeed not exist.
For us, they do.

[Atla]

Every member of the club sees exactly the same things.
How is that possible if all of us just invented them?
How do we manage to coordinate a common fictitious illusion so precisely?
So, for people who do not have the ability to see them, they do indeed not exist.
For us, they do.

Again, it would be really strange if living in the same part of the same physical universe with the same "laws", "constants" etc., in short given the same behaviour of the natural world around us, we would arrive at all kinds of different mathematics in our abstract thinking. That's why nominalism is most likely rational and platonism is most likely just magical thinking. We already have enough magical thinking on the planet, thank you.

The natural world around us seems to behave consistently all the way to the edge of the observable universe. That's how.

[godelian]

Every member of the club sees exactly the same things.
How is that possible if all of us just invented them?
How do we manage to coordinate a common fictitious illusion so precisely?
So, for people who do not have the ability to see them, they do indeed not exist.
For us, they do.

Again, it would be really strange if living in the same part of the same physical universe with the same "laws", "constants" etc., in short given the same behaviour of the natural world around us, we would arrive at all kinds of different mathematics in our abstract thinking. That's why nominalism is most likely rational and platonism is most likely just magical thinking. We already have enough magical thinking on the planet, thank you.

The natural world around us seems to behave consistently all the way to the edge of the observable universe. That's how.

Most mathematical objects do not or cannot even exist as physical objects. For example, where have you ever seen Cantor's aleph sequence? It would not even be possible because there are no actual infinities in the physical universe. Nonstandard numbers are also not observable in the physical universe. Not everybody will be able to observe them Platonically either. This is not about magical thinking but about talent. You perfectly well know that half the planet is completely inept in terms of mathematical ability. Do you really believe that they flunk their math exams just for the pleasure of flunking them? You know better than that!

[Atla]

Every member of the club sees exactly the same things.
How is that possible if all of us just invented them?
How do we manage to coordinate a common fictitious illusion so precisely?
So, for people who do not have the ability to see them, they do indeed not exist.
For us, they do.

Again, it would be really strange if living in the same part of the same physical universe with the same "laws", "constants" etc., in short given the same behaviour of the natural world around us, we would arrive at all kinds of different mathematics in our abstract thinking. That's why nominalism is most likely rational and platonism is most likely just magical thinking. We already have enough magical thinking on the planet, thank you.

The natural world around us seems to behave consistently all the way to the edge of the observable universe. That's how.

Most mathematical objects do not or cannot even exist as physical objects. For example, where have you ever seen Cantor's aleph sequence? It would not even be possible because there are no actual infinities in the physical universe. Nonstandard numbers are also not observable in the physical universe. Not everybody will be able to observe them Platonically either. This is not about magical thinking but about talent. You perfectly well know that half the planet is completely inept in terms of mathematical ability. Do you really believe that they flunk their math exams just for the pleasure of flunking them? You know better than that!

There are arguably no actual infinities in our physics, but we don't know if there are actual infinities in the physical universe.

The rest is just you not understanding that nominalism and platonism are philosophical interpretations, mathematics isn't inherently platonist and never was.

[godelian]

The rest is just you not understanding that nominalism and platonism are philosophical interpretations, mathematics isn't inherently platonist and never was.

Mathematics is multi-ontological. It is just string manipulation (formalism). This is always correct. It's just about template placeholders in a template structure (structuralism). That is also always correct. And so on. The only ontology that requires true "insight", is the Platonic one. Concerning nominalism, I don't know what to think about arbitrary abstractions. They don't ring a bell to me.

[Atla]

The only ontology that requires true "insight", is the Platonic one.

Like religion huh