If your branch of mathematics is so unapplied that you can’t even represent it in our universe, I suspect it’s no longer math.
Any maths can be represented the way it ususally is, by writing down some essentially aribtrary symbols. That
does not indicate anything about “correspondence” to reality. The problem is the “arbitrary” in arbitrary symbol.
Lets say space is three dimensional. You can write down a formula for 17 dimensional space, but that doens’t
mean you have a chunk of 17 dimesional space for the maths to correspond to. You just have chalk on a blackboard.
Sure. And yet, you can implement vectors in 17 dimensional spaces by writing down 17-dimensional vectors in row notation. Math predicts the outcome of operations on these entities.
Show me a 17-vector. And what is being predicited? The onlyy way to get at the behaviour is to do the math, and the only way to do the predictions is...to do the math. I think meaningful prediction requires some non-identity between predictee and predictor.
The predictions of the mathematics of 17-dimensional space would, yes, depend on the outcome of other operations such as addition and multiplication—operations we can implement more directly in matter.
I have personally relied on the geometry of 11-dimensional spaces for a particular curve-fitting model to produce reliable results. If, say, the Pythagorean theorem suddenly stopped applying above 3 dimensions, it simply would not have worked.
I’m seing pixels on a 2d screen. I’m not seeing an existing 17d dimensional thing.
The mathematics of 17d space predict the mathematics of 17d space. They couldn’t fail to. Which means no real prediction is happening at all. 1.0 is not a probability.
There are things we can model as 17-dimensional spaces, and when we do, the behavior comes out the way we were hoping. This is because of formal equivalence: the behavior of the numbers in a 17-dimensional vector space precisely corresponds to the geometric behavior of a counterfactual 17-dimensional euclidean space. You talk about one, you’re also saying something about the other.
There are things we can model as 17-dimensional spaces,
But they are not 17 dimensional spaces. They have different physics. Treating them as 17 dimesional isn’t modelling them because it isn’t representing thema as they are.
To be concrete, suppose we have a robotic arm with 17 degrees of freedom of movement. It’s current state can and should be represented as a 17-dimensional vector, to which you should do 17-dimensional math to figure out things like “Where is the robotic arm’s index finger pointing?” or “Can the robotic arm touch its own elbow?”
Not obvious. It would just be a redundant way of representing a 3d object in 3 space.
The point of contention is the claim that for any maths, there is something in reality for it to represent. Now, we can model a system of 10 particles as 1 particle in 30 dimensional space, but that doens;t prove that 30d maths
has something in reality to represent, since in reality there are 10 particles. Is was our decision, not reality’s to
treat is as 1 particle in a higher-d space.
Past a certain degree of complexity, there are lots of decisions about representing objects that are “ours, not reality’s”. For example, even if you represent the 10 particles as 10 vectors in 3D space, you still choose an origin, a scale, and a basis for 3D space, and all of these are arbitrary.
The 30-dimensional particle makes correct predictions of the behavior of the 10 particles. That should be enough.
Treaing a mathermatical formula as something that cranks out predictions is treating it as instrumentally, is treaing it unrealistically. But you cannot’ have coherent notion of modeling or representation if there is no real territory being modeled or represented.
To argue that all maths is representational, you either have to claim we are living in Tegmarks level IV, or you have to stretch the meaning of “representation” to meaninglessness. Kindly and Luke Sommers seem to be heading down the second route.
A correct 30D formula wll make correct predictions, Mathematical space also contains an infinity of formulations that are incorrect. Surely it is obvious that you can’t claim eveything in maths correctly models or predicts something in realiy.
Predicts something that could happen in reality (e.g. we’re not rejecting math with 2+2=4 apples just because I only have 3 apples in my kitchen), or
Is an abstraction of other mathematical ideas.
Do you claim that (2) is no longer modeling something in reality? It is arguably still predicting things about reality once you unpack all the layers of abstraction—hopefully at least it has consequences relevant to math that does model something.
Or do you think that I’ve missed a category in my description of math?
It is arguably still predicting things about reality once you unpack all the layers of abstraction
I don’t see what abstraction has to do with it. The Standard Model has about 18 parameters. Vary those, and it will mispredict. I don’t think all the infinity of incorrect variations of the SM are more abstract.
As a physicist, I can say with a moderate degree of authority: no.
I have seen mathematical equations to describe population genetics. That was not physics. I have seen mathematical equations used to describe supply and demand curves. That was not physics. Etc.
If you’re using math to model something, or even could so use it, that is sufficient for it to have a correspondent for purposes of the correspondence theory of truth.
If you’re using math to model something, or even could so use it, that is sufficient for it to have a correspondent for purposes of the correspondence theory of truth.
But that is not suffcient to show that all maths models.
… okay, you were confusing before, but now you’re exceptionally confusing. You’re saying that the standard model of particle physics is an example of math that doesn’t model anything?
Well, it doesn’t model our universe. And the Standard Model is awfully complicated for someone to build a condensed matter system implementing a randomized variant of it. But it’s still a quantum mechanical system, so I wouldn’t bet strongly against it.
And of course if someone decided for some reason to run a quantum simulation using this H-sm-random, then anything you mathematically proved about H-sm-random would be proved about the results of that simulation. The correspondence would be there between the symbols you put in and the symbols you got back, by way of the process used to generate those outputs. It just would be modeling something less cosmically grand than the universe itself, just stuff going on inside a computer. It wouldn’t be worth while to do… but it still corresponds to a relationship that would hold if you were dumb enough to go out of your way to bring it about.
The thing about the correspondence theory of truth is that once something has been reached as corresponding to something and thus being eligible to be true, it serves as a stepping-stone to other things. You don’t need to work your way all the way down to ‘ground floor’ in one leap. You’re allowed to take general cases, not all of which need to be instantiated. Correspondence to patterns instead of instances is a thing.
And of course if someone decided for some reason to run a quantum simulation using this H-sm-random, the anything you mathematically proved about H-sm-random would be proved about the results of that simulation.
Which, as in your other examples, is case of a model modeling a model. You can build something physical
that simulates a universe where electrons have twice the mass, and you can predict the virtual behaviour
of the simulation with an SM where the electron mass paramter is doubled, but the simulation will be made
of electrons with standard mass.
The correspondence would be there between the symbols you put in and the symbols you got back, by way of the process used to generate those outputs. It just would be modeling something less cosmically grand than the universe itself, just stuff going on inside a computer.
It wouldn’t be modeliing reality.
The thing about the correspondence theory of truth
..is that it is a poor fit for mathematical truth. You are making mathetmatical theorems correspondnce-true
by giving them something artificial to correspond to. Before the creation of a simulaiton at time T, there
is nothing for them to correspond to.This is a mismatch with the intuition that mathematical truths are timelessly true.
is that once something has been reached as corresponding to something and thus being eligible to be true, it serves as a stepping-stone to other things. You don’t need to work your way all the way down to ‘ground floor’ in one leap. You’re allowed to take general cases, not all of which need to be instantiated. Correspondence to patterns instead of instances is a thing.
You can gerrymander CToT into something that works, however inelegantly, for maths, or you can abandon it in favour something that doesn’t need gerrymandering.
Any maths can be represented the way it ususally is, by writing down some essentially aribtrary symbols. That does not indicate anything about “correspondence” to reality. The problem is the “arbitrary” in arbitrary symbol.
Lets say space is three dimensional. You can write down a formula for 17 dimensional space, but that doens’t mean you have a chunk of 17 dimesional space for the maths to correspond to. You just have chalk on a blackboard.
Sure. And yet, you can implement vectors in 17 dimensional spaces by writing down 17-dimensional vectors in row notation. Math predicts the outcome of operations on these entities.
Show me a 17-vector. And what is being predicited? The onlyy way to get at the behaviour is to do the math, and the only way to do the predictions is...to do the math. I think meaningful prediction requires some non-identity between predictee and predictor.
(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6) is a perfectly valid 17-vector.
The predictions of the mathematics of 17-dimensional space would, yes, depend on the outcome of other operations such as addition and multiplication—operations we can implement more directly in matter.
I have personally relied on the geometry of 11-dimensional spaces for a particular curve-fitting model to produce reliable results. If, say, the Pythagorean theorem suddenly stopped applying above 3 dimensions, it simply would not have worked.
I’m seing pixels on a 2d screen. I’m not seeing an existing 17d dimensional thing.
The mathematics of 17d space predict the mathematics of 17d space. They couldn’t fail to. Which means no real prediction is happening at all. 1.0 is not a probability.
There are things we can model as 17-dimensional spaces, and when we do, the behavior comes out the way we were hoping. This is because of formal equivalence: the behavior of the numbers in a 17-dimensional vector space precisely corresponds to the geometric behavior of a counterfactual 17-dimensional euclidean space. You talk about one, you’re also saying something about the other.
Is this point confusing to you?
But they are not 17 dimensional spaces. They have different physics. Treating them as 17 dimesional isn’t modelling them because it isn’t representing thema as they are.
To be concrete, suppose we have a robotic arm with 17 degrees of freedom of movement. It’s current state can and should be represented as a 17-dimensional vector, to which you should do 17-dimensional math to figure out things like “Where is the robotic arm’s index finger pointing?” or “Can the robotic arm touch its own elbow?”
Not obvious. It would just be a redundant way of representing a 3d object in 3 space.
The point of contention is the claim that for any maths, there is something in reality for it to represent. Now, we can model a system of 10 particles as 1 particle in 30 dimensional space, but that doens;t prove that 30d maths has something in reality to represent, since in reality there are 10 particles. Is was our decision, not reality’s to treat is as 1 particle in a higher-d space.
Past a certain degree of complexity, there are lots of decisions about representing objects that are “ours, not reality’s”. For example, even if you represent the 10 particles as 10 vectors in 3D space, you still choose an origin, a scale, and a basis for 3D space, and all of these are arbitrary.
The 30-dimensional particle makes correct predictions of the behavior of the 10 particles. That should be enough.
Treaing a mathermatical formula as something that cranks out predictions is treating it as instrumentally, is treaing it unrealistically. But you cannot’ have coherent notion of modeling or representation if there is no real territory being modeled or represented.
To argue that all maths is representational, you either have to claim we are living in Tegmarks level IV, or you have to stretch the meaning of “representation” to meaninglessness. Kindly and Luke Sommers seem to be heading down the second route.
A correct 30D formula wll make correct predictions, Mathematical space also contains an infinity of formulations that are incorrect. Surely it is obvious that you can’t claim eveything in maths correctly models or predicts something in realiy.
I’d say that math either
Predicts something that could happen in reality (e.g. we’re not rejecting math with 2+2=4 apples just because I only have 3 apples in my kitchen), or
Is an abstraction of other mathematical ideas.
Do you claim that (2) is no longer modeling something in reality? It is arguably still predicting things about reality once you unpack all the layers of abstraction—hopefully at least it has consequences relevant to math that does model something.
Or do you think that I’ve missed a category in my description of math?
I don’t see what abstraction has to do with it. The Standard Model has about 18 parameters. Vary those, and it will mispredict. I don’t think all the infinity of incorrect variations of the SM are more abstract.
Who said vector spaces have anything to do with physics? That’s not math anymore, that’s physics.
Using math to model reality is physics. Phsycis doens’t use all of math, so some math doesn’ model anything real.
As a physicist, I can say with a moderate degree of authority: no.
I have seen mathematical equations to describe population genetics. That was not physics. I have seen mathematical equations used to describe supply and demand curves. That was not physics. Etc.
If you’re using math to model something, or even could so use it, that is sufficient for it to have a correspondent for purposes of the correspondence theory of truth.
But that is not suffcient to show that all maths models.
Well, you can use math for something other than modeling, sure. Can you give a more concrete example of some math you claim doesn’t model anything?
The Standard Model with its 18 parameters set to random values.
… okay, you were confusing before, but now you’re exceptionally confusing. You’re saying that the standard model of particle physics is an example of math that doesn’t model anything?
No, I am saying a mutated, deviant form doens’t model anything -- “with its 18 parameters set to random values”.
Well, it doesn’t model our universe. And the Standard Model is awfully complicated for someone to build a condensed matter system implementing a randomized variant of it. But it’s still a quantum mechanical system, so I wouldn’t bet strongly against it.
And of course if someone decided for some reason to run a quantum simulation using this H-sm-random, then anything you mathematically proved about H-sm-random would be proved about the results of that simulation. The correspondence would be there between the symbols you put in and the symbols you got back, by way of the process used to generate those outputs. It just would be modeling something less cosmically grand than the universe itself, just stuff going on inside a computer. It wouldn’t be worth while to do… but it still corresponds to a relationship that would hold if you were dumb enough to go out of your way to bring it about.
The thing about the correspondence theory of truth is that once something has been reached as corresponding to something and thus being eligible to be true, it serves as a stepping-stone to other things. You don’t need to work your way all the way down to ‘ground floor’ in one leap. You’re allowed to take general cases, not all of which need to be instantiated. Correspondence to patterns instead of instances is a thing.
Which, as in your other examples, is case of a model modeling a model. You can build something physical that simulates a universe where electrons have twice the mass, and you can predict the virtual behaviour of the simulation with an SM where the electron mass paramter is doubled, but the simulation will be made of electrons with standard mass.
It wouldn’t be modeliing reality.
..is that it is a poor fit for mathematical truth. You are making mathetmatical theorems correspondnce-true by giving them something artificial to correspond to. Before the creation of a simulaiton at time T, there is nothing for them to correspond to.This is a mismatch with the intuition that mathematical truths are timelessly true.
You can gerrymander CToT into something that works, however inelegantly, for maths, or you can abandon it in favour something that doesn’t need gerrymandering.
It’s not gerrymandering. What you are doing is gerrymandering. Picking and choosing which parts of the territory we are and aren’t allowed to model.
The territory includes the map.
But not as a map. Maphood is in the eye of the beholder.
The eye of the beholder is part of the territory too. It is a matter of fact that it takes that part of the territory to be a map.
Maphood is still not a matter of fact about maps.