Tarski Undefinability Theorem Succinctly Refuted

What is the basis for reason? And mathematics?

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Re: Tarski Undefinability Theorem Succinctly Refuted

Post by Speakpigeon » Thu Apr 18, 2019 8:13 am

Physicists are perfectly capable of creating whatever maths they need and they have to because oftentimes no mathematical theory is available.
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Re: Tarski Undefinability Theorem Succinctly Refuted

Post by Arising_uk » Thu Apr 18, 2019 8:17 am

Which Kant pretty much pointed out over 200 years ago.
Critique of Pure Reason

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Re: Tarski Undefinability Theorem Reexamined

Post by Arising_uk » Thu Apr 18, 2019 8:25 am

Logik wrote: Not at all. Park physics aside for a second. Language (logos, logic) is a tool for self-expression. ...
If you mean communication then I'd agree.
A tool for describing/narrating your own experiences. So if you can construct poetry with English, you can construct poetry with Mathematics.
You just need good command and understanding of your tools. And (the important bit) no rules! ...
Currently I doubt a natural language can be reduced to mathematics.
As Richard Feynman said: What I cannot create I do not understand.

Computer scientists create formal languages.

What experimental physicists have been doing for ever and ever is run to Mathematicians and ask "Do you have anything that can help here?".
And what mathematicians have been bitching about is how Physicists bastardize their "beautiful mathematics". ...
The few theoretical physicists I've met don't really run to mathematicians as they have the mathematical ability and tools they need, they're just not mathematicians in the sense of applying maths to itself.
What boggles my mind is that physicists can't construct the Mathematics they need from first principles! What? Just bloody describe what you experienced!
But they do?
Express what happened in a formal language. Program dammit ! I side with physicists here. Screw "beauty" - if it's stupid an it works it isn't stupid.
Is Maths not a formal language?

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Re: Tarski Undefinability Theorem Succinctly Refuted

Post by Speakpigeon » Thu Apr 18, 2019 8:31 am

Prove
3. Mathematics a. To demonstrate the validity of (a hypothesis or proposition).
Demonstrate
Word origin of 'demonstrate' C16: from Latin dēmonstrāre to point out, from monstrāre to show
So, for logicians and mathematicians, to prove a proposition p is to show the validity of p.
Validity
4. Logic b. Correctly inferred or deduced from a premise: a valid conclusion.
Proof relies on logical rules of inference.
The Modus Ponens is usually the basic rule for such demonstrations.
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Re: Tarski Undefinability Theorem Reexamined

Post by Logik » Thu Apr 18, 2019 8:59 am

Arising_uk wrote:
Thu Apr 18, 2019 8:25 am
If you mean communication then I'd agree.
Not at first. Ultimately communication, but initially you need to build a working model. For your own sake.

You need to capture your own experiences and synthesize them into "knowledge". A howto. An algorithm.

Once it works, then you can communicate it.
Arising_uk wrote:
Thu Apr 18, 2019 8:25 am
Currently I doubt a natural language can be reduced to mathematics.
But that's not how Physics works. You are never reducing natural language to mathematics - you are reducing your experiences to a mathematical model. The mathematical model forces us to invent new natural language. But only after we actually have a working model.

Truth is conceptual first...
Arising_uk wrote:
Thu Apr 18, 2019 8:25 am
Is Maths not a formal language?
Yes, but Mathematicians don't see mathematics as an instrument. They focus on the the Mathematical ideals. Beauty, aesthetics, consistency.
It goes a little stray into being too abstract at times. It loses contact with the ground and it becomes idealism like all other.

I am not allowed to lose contact with the ground. Because computers are the ground.

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Re: Tarski Undefinability Theorem Succinctly Refuted

Post by Logik » Thu Apr 18, 2019 9:02 am

Arising_uk wrote:
Thu Apr 18, 2019 8:17 am
Which Kant pretty much pointed out over 200 years ago.
Critique of Pure Reason
That's just the way you have indexed the origin of the idea in your head.
It's older than Kant. He's just the most popular philosopher to have endorsed it.

It doesn't really matter - it's not about giving credit, it's about becoming aware of such things.
There aren't all that many new ideas. They are just old ideas in new language.

There is no such thing as a new idea. It is impossible. We simply take a lot of old ideas and put them into a sort of mental kaleidoscope. We give them a turn and they make new and curious combinations. We keep on turning and making new combinations indefinitely; but they are the same old pieces of colored glass that have been in use through all the ages --Mark Twain

The phenomenology of human experience hasn't changed much in 5000 years, even though "phenomenology" itself can't be traced further back than 2-300 years.

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Re: Tarski Undefinability Theorem Succinctly Refuted

Post by Logik » Thu Apr 18, 2019 9:37 am

Speakpigeon wrote:
Thu Apr 18, 2019 8:13 am
Physicists are perfectly capable of creating whatever maths they need and they have to because oftentimes no mathematical theory is available.
EB
That's not true. For the same reason you keep referring to dictionaries rather than inventing your own words to describe new ideas.

When it comes to natural languages a made up word (one that nobody else uses) is a meaningless word. This is Wittgenstein's private language argument.

Physicists are empiricists, not linguists. When doing physics they don't have to explain their concepts to anybody but themselves initially.

That private language argument doesn't apply to metalanguages (formal systems).
A made up concept is meaningful if I can understand it and if it represents the structure of my own experience of whatever it is that took place.
And evidence for its meaning is the fact that I can explain it to my computer; to build a model; which predicts correctly.

You could say that I am making an argument for structuralism as complementary to deconstruction. Which wouldn't be far from the truth since analysis <-> synthesis are complementary in complexity/systems theory.

Ultimately though. Truth is holistic/synthetic and semantic. That's why all philosophers are collectively and approximately right,

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Re: Tarski Undefinability Theorem Reexamined

Post by wtf » Wed Apr 24, 2019 4:16 am

Logik wrote:
Sat Apr 06, 2019 11:19 pm
You are mistaken Conceptually the fields of computation, physics and mathematics are isomorphic.
You know anything about Cartesian closed categories? I dabble in a little category theory and watched a video with Steve Awodey describing how the lambda calculus can be interpreted as a Cartesian closed category. So I think I understand where you're coming from on this point. The claim that physics is a computation is a metaphysical speculation without evidentiary basis. Also, plenty of important categories aren't Cartesian closed: Topological spaces, for one; and smooth manifolds, for another. The latter are essential to physics.

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

Also see exponential objects. These are basically an abstraction of currying.

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

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Re: Tarski Undefinability Theorem Reexamined

Post by Logik » Thu Apr 25, 2019 8:34 am

wtf wrote:
Wed Apr 24, 2019 4:16 am
Logik wrote:
Sat Apr 06, 2019 11:19 pm
You are mistaken Conceptually the fields of computation, physics and mathematics are isomorphic.
You know anything about Cartesian closed categories? I dabble in a little category theory and watched a video with Steve Awodey describing how the lambda calculus can be interpreted as a Cartesian closed category. So I think I understand where you're coming from on this point. The claim that physics is a computation is a metaphysical speculation without evidentiary basis. Also, plenty of important categories aren't Cartesian closed: Topological spaces, for one; and smooth manifolds, for another. The latter are essential to physics.

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

Also see exponential objects. These are basically an abstraction of currying.

https://en.wikipedia.org/wiki/Exponential_object
I think I mentioned before that I come from a strictly computational background, and in particular - the applied, not the theoretical side.
So most of what I am doing presently is attempting to connect the dots between my intuitions and tacit knowledge back to the theory. So that I can re-calibrate my intuitions towards the quantum-computational paradigm.

To say that it's metaphysical speculation without evidentiary basis tells me that your own metacognition lacks a self-referential model of what evidentiary testing is and how it works. Obviously no amount of brain surgery is going to extract the evidence-meter from our heads, so we are stuck with Platonism. Because I subscribe to a digital ontology (information is foundational) to me the words "evidence" and "information" are synonymous. In my meta-cognition evidentiary testing is approximately Bayesian inference.

Is it metaphysical speculation? Yeah. But it works. It's a pragmatic truth and in that regard it's self-fulfilling.


Where I have landed presently is that I have internalized the BHK Interpretation way before I knew what it's called, and because I am trying to disconfirm my own bias of "all proofs are algorithms" - although it's kinda difficult to get there when you axiomatically start with "everything is information".

If I understand your concern re: Cartesian closed spaces, it's basically the same concern all of Quantum Mechanics have been complaining about. Infinities break the mathematics - lets renormalise everything! And renormalization is effectively turning everything into a Cartesian closed space. Which is Homotopy's H:X * [0,1] -> Y

It's pretty much why I am subscribed to ultrafinitism, but (and this is the big BUT) if QM ends up towards an interpretation which favours global hidden variables it will all blow up, and exactly for the reasons you point out. Quantum fields may not be Cartesian closed.

But, of course - I am still making my way through the HoTT book so that I can translate my thoughts from "Computer Scientist" to "Mathematician".
Last edited by Logik on Thu Apr 25, 2019 12:20 pm, edited 2 times in total.

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Re: Tarski Undefinability Theorem Reexamined

Post by Logik » Thu Apr 25, 2019 9:15 am

Logik wrote:
Thu Apr 25, 2019 8:34 am
... plenty of important categories aren't Cartesian closed: Topological spaces, for one; and smooth manifolds, for another. The latter are essential to physics.

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

Also see exponential objects. These are basically an abstraction of currying.

https://en.wikipedia.org/wiki/Exponential_object
P.S Thanks! This is exactly where I need to be looking.

I need to understand the limits of differentiable manifolds.

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Re: Tarski Undefinability Theorem Reexamined

Post by wtf » Fri Apr 26, 2019 12:01 am

Logik wrote:
Thu Apr 25, 2019 8:34 am
If I understand your concern re: Cartesian closed spaces, it's basically the same concern all of Quantum Mechanics have been complaining about. Infinities break the mathematics - lets renormalise everything! And renormalization is effectively turning everything into a Cartesian closed space. Which is Homotopy's H:X * [0,1] -> Y
I'm talking about category theory. It's pretty abstract. If you've seen any abstract algebra (groups, rings, fields) then you can think of category theory as "abstract abstract algebra." It's a structural way of thinking about mathematics that was invented in the 1940's. In recent decades it's started to permeate computer science, geometry, and logic.

I saw some of it it math school a long time ago and have recently been diving back in. I ran across this concept of Cartesian closed categories, and it turns out that they are a way of expressing lambda calculus and intuitionistic logic and a whole bunch of other related stuff. So I was just asking if you happen to have run across them.

Haskell is based on the same set of ideas, and in fact there are a lot of Youtube videos on category theory that are being done by Haskell programmers and other functional programmers.

It turns out that logic, geometry, math, and even some parts of physics are being subsumed into category theory these days. It's the magic way of thinking that sort of integrates everything. But I was just asking if you've come across it. It's definitely obscure in the scheme of things, not yet mainstream in any sense.

The idea of an exponential object, which I linked, is like a huge abstractification of the idea of Currying functions. That's another point of contact between programming and category theory.

ps -- Actually HOTT is the other new alternative foundation. That's something I'd like to know more about. I know what homotopies are from a mathematical perspective but I haven't looked at HOTT much except for a couple of videos and websites and such.

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Re: Tarski Undefinability Theorem Reexamined

Post by Logik » Sun Apr 28, 2019 12:21 pm

wtf wrote:
Fri Apr 26, 2019 12:01 am
I'm talking about category theory. It's pretty abstract. If you've seen any abstract algebra (groups, rings, fields) then you can think of category theory as "abstract abstract algebra." It's a structural way of thinking about mathematics that was invented in the 1940's. In recent decades it's started to permeate computer science, geometry, and logic.

I saw some of it it math school a long time ago and have recently been diving back in. I ran across this concept of Cartesian closed categories, and it turns out that they are a way of expressing lambda calculus and intuitionistic logic and a whole bunch of other related stuff. So I was just asking if you happen to have run across them.
I have run across them in as much as I use them every day of my life.

When I said that the divergence in our thought-processes comes from nomenclature and primitives. I had category theory in mind.

My primitive is the Black box. A black box is a function from category theory.

wtf wrote:
Fri Apr 26, 2019 12:01 am
Haskell is based on the same set of ideas, and in fact there are a lot of Youtube videos on category theory that are being done by Haskell programmers and other functional programmers.
Yep. A function is a black box.
wtf wrote:
Fri Apr 26, 2019 12:01 am
It turns out that logic, geometry, math, and even some parts of physics are being subsumed into category theory these days. It's the magic way of thinking that sort of integrates everything. But I was just asking if you've come across it. It's definitely obscure in the scheme of things, not yet mainstream in any sense.

The idea of an exponential object, which I linked, is like a huge abstractification of the idea of Currying functions. That's another point of contact between programming and category theory.
Indeed "currying" is just functional decomposition. Reduction even: https://en.wikipedia.org/wiki/Reduction_(complexity)

Category theory is pretty much how I think about the world. Start with highest level of abstraction, then decompose down to least complex elements.

This is the analytic/synthetic continuum in complexity/systems theory. Systems. Systems of systems. Systems of systems of systems.

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Re: Tarski Undefinability Theorem Reexamined

Post by wtf » Tue Apr 30, 2019 11:40 pm

Logik wrote:
Sun Apr 28, 2019 12:21 pm
Category theory is pretty much how I think about the world. Start with highest level of abstraction, then decompose down to least complex elements.
What do you think about universal mapping properties?

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Re: Tarski Undefinability Theorem Reexamined

Post by Logik » Wed May 01, 2019 12:02 am

wtf wrote:
Tue Apr 30, 2019 11:40 pm
What do you think about universal mapping properties?
It's a formalization of Equifinality.

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

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Re: Tarski Undefinability Theorem Reexamined

Post by wtf » Wed May 01, 2019 12:46 am

Logik wrote:
Wed May 01, 2019 12:02 am
wtf wrote:
Tue Apr 30, 2019 11:40 pm
What do you think about universal mapping properties?
It's a formalization of Equifinality.

https://en.wikipedia.org/wiki/Equifinality
You have not convinced me that when we talk about category theory we are talking about the same thing.

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