Simplest possible notion of a formal system

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Simplest possible notion of a formal system
https://plato.stanford.edu/entries/analyticsynthetic/
When one properly makes the analytic versus synthetic distinction one realizes that
the entire body of conceptual knowledge
is entirely comprised of stipulated relations between expressions of language.
This explicitly includes but is not limited to every single detail about every aspect of every mathematical system along with the semantics of every mathematical expression of any of these systems.
The only possible rebuttal to the above claim is to find a valid counterexample of a pure concept that is not entirely defined using language. As long as we stay on the analytic side of the analytic/synthetic distinction this is impossible.
To formalize the body of conceptual knowledge merely requires expressing the stipulated relations between expressions of language as relations between finite strings.
So we end up with the simplest possible notion of a formal system that can express every element of the entire body of conceptual knowledge as simply a set of stipulated relations between finite strings.
The notion of True(x) is simply the satisfaction of the stipulated relations in the body of conceptual knowledge and the notion of False(x) is the satisfaction of the negation of x in this same body of knowledge.
To understand this idea think of this simplification of Curry/Howard Correspondence.
A software function is named according to the Relation that it represents and returns Boolean.
Finite strings are always Unicode characters. Arguments to this function are finite strings.
">"("5","3") returns true
">"("3","5") returns false
">"("orange","apple") returns false // type mismatch error
">"("apple","orange") returns false // type mismatch error
"◁" is a type of operator
"◁"("cats", "animals") // returns true
Copyright 2019 Pete Olcott
When one properly makes the analytic versus synthetic distinction one realizes that
the entire body of conceptual knowledge
is entirely comprised of stipulated relations between expressions of language.
This explicitly includes but is not limited to every single detail about every aspect of every mathematical system along with the semantics of every mathematical expression of any of these systems.
The only possible rebuttal to the above claim is to find a valid counterexample of a pure concept that is not entirely defined using language. As long as we stay on the analytic side of the analytic/synthetic distinction this is impossible.
To formalize the body of conceptual knowledge merely requires expressing the stipulated relations between expressions of language as relations between finite strings.
So we end up with the simplest possible notion of a formal system that can express every element of the entire body of conceptual knowledge as simply a set of stipulated relations between finite strings.
The notion of True(x) is simply the satisfaction of the stipulated relations in the body of conceptual knowledge and the notion of False(x) is the satisfaction of the negation of x in this same body of knowledge.
To understand this idea think of this simplification of Curry/Howard Correspondence.
A software function is named according to the Relation that it represents and returns Boolean.
Finite strings are always Unicode characters. Arguments to this function are finite strings.
">"("5","3") returns true
">"("3","5") returns false
">"("orange","apple") returns false // type mismatch error
">"("apple","orange") returns false // type mismatch error
"◁" is a type of operator
"◁"("cats", "animals") // returns true
Copyright 2019 Pete Olcott
Last edited by PeteOlcott on Mon Aug 12, 2019 6:13 pm, edited 5 times in total.
Re: Simplest possible notion of a formal system
Curses! I was just going to steal this and publish it as my own.
Ok Pete, seriously, I cannot distinguish your system from what's known as True Arithmetic, the set of all true statements about the integers. But I've seen you deny (on Reddit I believe) that your system is TA. Can you explain the distinction?
https://en.wikipedia.org/wiki/True_arithmetic

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 Joined: Mon Jul 25, 2016 6:55 pm
Re: Simplest possible notion of a formal system
When I refer to the entire body of conceptual knowledge why the Hell would you think that I am only talking about arithmetic?
Re: Simplest possible notion of a formal system
Is the restriction of your system with to arithmetic of the natural numbers the same as TA then?PeteOlcott wrote: ↑Fri Aug 09, 2019 8:01 pmWhen I refer to the entire body of conceptual knowledge why the Hell would you think that I am only talking about arithmetic?
And as far as why I'd ask. YOU are the one talking about the Halting problem and Gödel's incompleteness theorems. It's perfectly reasonable to assume you are talking about mathematics.

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Re: Simplest possible notion of a formal system
I am not talking about numbers. I am talking about anything that can possibly be said by anyone about anything.
Re: Simplest possible notion of a formal system
PeteOlcott wrote: ↑Fri Aug 09, 2019 8:28 pmI am not talking about numbers. I am talking about anything that can possibly be said by anyone about anything.
My mistake Pete, I didn't realize your cat typed that by walking on your keyboard. It was wrong of me to ascribe those words to you personally. I now realize that when someone using your handle said that your theory applies to mathematics, you did NOT mean your theory applies to mathematics.PeteOlcott wrote: ↑Fri Aug 09, 2019 6:36 pmhttps://plato.stanford.edu/entries/analyticsynthetic/
This explicitly includes every single detail about every aspect of every mathematical system along with the semantics of every mathematical expression of any of these systems.

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Re: Simplest possible notion of a formal system
First of all the body of conceptual knowledge contains everything that anyone could ever say about anything in any language whatsoever.
Once you understand this, then (AND ONLY THEN) do we move to further elaborate that a tiny subset of this is everything that anyone could ever say about mathematics in any language whatsoever.
If you do not see what I say in necessary prerequisite order it cannot possibly be sufficiently understood.
This is my life's work not any mere entertainment hobby.
Once you understand this, then (AND ONLY THEN) do we move to further elaborate that a tiny subset of this is everything that anyone could ever say about mathematics in any language whatsoever.
If you do not see what I say in necessary prerequisite order it cannot possibly be sufficiently understood.
This is my life's work not any mere entertainment hobby.
Re: Simplest possible notion of a formal system
Ok fine. So when someone using your handle wrote:PeteOlcott wrote: ↑Fri Aug 09, 2019 8:46 pmFirst of all the body of conceptual knowledge contains everything that anyone could ever say about anything in any language whatsoever.
Once you understand this, then (AND ONLY THEN) do we move to further elaborate that a tiny subset of this is everything that anyone could ever say about mathematics in any language whatsoever.
If you do not see what I say in necessary prerequisite order it cannot possibly be sufficiently understood.
This is my life's work not any mere entertainment hobby.
"This explicitly includes every single detail about every aspect of every mathematical system along with the semantics of every mathematical expression of any of these systems."
was that someone other than you? Perhaps you should change your password.
If it was in fact you who wrote those words, I am asking you, just as someone else asked you on Reddit, whether your system restricted to the natural numbers is the same as True Arithmetic.
The reason people ask this is to try to understand what you are saying.

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Re: Simplest possible notion of a formal system
It seems that you are intentionally playing head games and are not interested in any honest dialogue.
Re: Simplest possible notion of a formal system
You are overstating your case, which ONLY applies to analytic knowledge as synthetic knowledge has to include the empiricalPeteOlcott wrote: ↑Fri Aug 09, 2019 6:36 pmhttps://plato.stanford.edu/entries/analyticsynthetic/
When one properly makes the analytic versus synthetic distinction one realizes that
the entire body of conceptual knowledge
is entirely comprised of stipulated relations between expressions of language.
This explicitly includes but is not limited to every single detail about every aspect of every mathematical system along with the semantics of every mathematical expression of any of these systems.
The only possible rebuttal to the above claim is to find a valid counterexample of a pure concept that is not entirely defined using language. As long as we stay on the analytic side of the analytic/synthetic distinction this is impossible.
To formalize the body of conceptual knowledge merely requires expressing the stipulated relations between expressions of language as relations between finite strings.
So we end up with the simplest possible notion of a formal system that can express every element of the entire body of conceptual knowledge as simply a set of stipulated relations between finite strings.
The notion of True(x) is simply the satisfaction of the stipulated relations in the body of conceptual knowledge and the notion of False(x) is the satisfaction of the negation of x in this same body of knowledge.
Copyright 2019 Pete Olcott

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 Joined: Mon Jul 25, 2016 6:55 pm
Re: Simplest possible notion of a formal system
Precisely limited my case to the analytic side of the analytic / synthetic distinction is not overstating my case at all. Do you know what a concept is? Is it analytic or synthetic?
 RCSaunders
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Re: Simplest possible notion of a formal system
"Everything that anyone could ever say about anything in any language whatsoever," is not knowledge. Some of it might be knowledge, but most of it would be just wrong or nonsense.PeteOlcott wrote: ↑Fri Aug 09, 2019 8:46 pmFirst of all the body of conceptual knowledge contains everything that anyone could ever say about anything in any language whatsoever.
This sounds like a very basic epistemological mistake about the nature of knowledge.
Re: Simplest possible notion of a formal system
Yes, here is my rebutal:PeteOlcott wrote: ↑Fri Aug 09, 2019 6:36 pmhttps://plato.stanford.edu/entries/analyticsynthetic/
When one properly makes the analytic versus synthetic distinction one realizes that
the entire body of conceptual knowledge
is entirely comprised of stipulated relations between expressions of language.
This explicitly includes but is not limited to every single detail about every aspect of every mathematical system along with the semantics of every mathematical expression of any of these systems.
The only possible rebuttal to the above claim is to find a valid counterexample of a pure concept that is not entirely defined using language. As long as we stay on the analytic side of the analytic/synthetic distinction this is impossible.
To formalize the body of conceptual knowledge merely requires expressing the stipulated relations between expressions of language as relations between finite strings.
So we end up with the simplest possible notion of a formal system that can express every element of the entire body of conceptual knowledge as simply a set of stipulated relations between finite strings.
The notion of True(x) is simply the satisfaction of the stipulated relations in the body of conceptual knowledge and the notion of False(x) is the satisfaction of the negation of x in this same body of knowledge.
Copyright 2019 Pete Olcott
⊙
I am assuming I will have to explain it further.
But besides that I agree with everything else you said.
 RCSaunders
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Re: Simplest possible notion of a formal system
A concept is neither. Even if the socalled analytic/synthetic dichotomy were true (it isn't), it would have nothing to do with concepts. I do not think you know what a concept is.PeteOlcott wrote: ↑Sat Aug 10, 2019 12:50 amDo you know what a concept is? Is it analytic or synthetic?
So, from, "Epistemology, Concepts:"
The Purpose Of Concepts
Words, in any language, represent concepts. Concepts have a single function which is to identify existents. Existents are anything that exists, ontologically (materially), or epistemologically. Material existents (entities) and epistemological existents are all that exists.
The Structure Of Concepts
"A concept consists of two components a "perceivable existent," and a "specification." The, "perceivable existent," is a symbol, usually a spoken or written word. The "specification" is a definition which specifies or indicates the existent or existents the concept identifies.
"The word (or other perceivable symbol) for a concept is not the concept. The word is our means of being conscious of the concept. The concept is the identification of an existent. The definition of a concept indicates what existent a concept identifies."
You'll have to read the rest of the article to truly understand concepts.
Re: Simplest possible notion of a formal system
Both.PeteOlcott wrote: ↑Sat Aug 10, 2019 12:50 amPrecisely limited my case to the analytic side of the analytic / synthetic distinction is not overstating my case at all. Do you know what a concept is? Is it analytic or synthetic?
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