## Conceptual Truth can be understood as math

What is the basis for reason? And mathematics?

Moderators: AMod, iMod

PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Truth can be understood as math

Eodnhoj7 wrote: Wed Aug 14, 2019 3:09 pm
PeteOlcott wrote: Wed Aug 14, 2019 2:33 am
Eodnhoj7 wrote: Wed Aug 14, 2019 2:03 am
Yes, basic counting. Anything can be quantified, thus we may observe how a phenomenon exists in time and space, but we never really know its qualities.

The reverse works for qualifying a phenomenon into categories. I may qualify a rose but it does not take into account all the other types of flowers. Or I may qualifies flowers, but it does not take into account other organisms. The same applies for organisms and minerals etc.

Thus we are left with a paradox in the act of measurement and if any unity or synthesis is to be observed we have to look at common denominators.
I would estimate that you are using the term paradox incorrectly.

A paradox is a statement that, despite apparently valid reasoning from true premises, leads to an apparently-self-contradictory or logically unacceptable conclusion.
False, it is correct.

In quantifying a phenomenon, the quantifier can mean an infinite number of things. However the quantifer only exists if something can be counted. This requires a quality. Thus in becoming more precise in a quantity, we become less precise in a quality. In counting oranges we lose detail about the qualities of the orange.

The reverse occurs.

In becoming defined in one thing, you become undefined in another. Increasing analysis results in increasing obscurity.
Does at least one existence exist?
Does justice ever exist? How many "justice" exist?
Skepdick
Posts: 4760
Joined: Fri Jun 14, 2019 11:16 am

### Re: Truth can be understood as math

PeteOlcott wrote: Wed Aug 14, 2019 5:38 pm You can form any logical incoherence that you want. the high level abstract notion of
a formal system from which every other formal system inherits calls out all logical
incoherence.
Oh really! Can you define the abstract class/type for a "logical system" ?
And I can instantiate that?

Seems you are preaching for convention over configuration.

Is there no lengths to which you wouldn't go to in order to take expressive power and flexibility away from me?
Last edited by Skepdick on Wed Aug 14, 2019 5:57 pm, edited 1 time in total.
Skepdick
Posts: 4760
Joined: Fri Jun 14, 2019 11:16 am

### Re: Truth can be understood as math

PeteOlcott wrote: Wed Aug 14, 2019 5:43 pm The simplest measure of logical incoherence is contradiction.
Every self contradictory expression of language must be rejected as logically incoherent:
"This sentence is not true"
"This sentence is not provable".
Must both be rejected.
So the Ruby program I just gave you is incoherent?

OK... Then how does the computer understand it?
PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Truth can be understood as math

Skepdick wrote: Wed Aug 14, 2019 5:54 pm
PeteOlcott wrote: Wed Aug 14, 2019 5:38 pm You can form any logical incoherence that you want. the high level abstract notion of
a formal system from which every other formal system inherits calls out all logical
incoherence.
Oh really! Can you define the abstract class/type for a "logical system" ?
finite-string_01 [is_stipulated_to_be_of_type] finite-string_02
Skepdick
Posts: 4760
Joined: Fri Jun 14, 2019 11:16 am

### Re: Truth can be understood as math

PeteOlcott wrote: Wed Aug 14, 2019 5:57 pm finite-string_01 [is_stipulated_to_be_of_type] finite-string_02
The Ruby program I have offered you meets this criterion.

It halts. Obviously it's finite.
PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Truth can be understood as math

Skepdick wrote: Wed Aug 14, 2019 5:58 pm
PeteOlcott wrote: Wed Aug 14, 2019 5:57 pm finite-string_01 [is_stipulated_to_be_of_type] finite-string_02
The Ruby program I have offered you meets this criterion.

It halts. Obviously it's finite.
I am not going to learn ruby to replay to one post.
The best that I can tell from your replay is that the Ruby program proves that I am correct.

Generically the high level abstraction of a formal system that I am referring to works this way:
finite_string_X [is_assigned_the_stipulated_relation_Y] to finite_string_Z
Skepdick
Posts: 4760
Joined: Fri Jun 14, 2019 11:16 am

### Re: Truth can be understood as math

PeteOlcott wrote: Wed Aug 14, 2019 6:02 pm I am not going to learn ruby to replay to one post.
You don't have to. Ruby is Turing-complete. All Turing-complete languages are FUNCTIONALLY EQUIVALENT.
Therefore there exists some_function() which can translate the Ruby logic into byte-code, or Assembly language, or Boolean logic, or Lambda calculus.

That's the definition of an interpreter/compiler. Right?!?
PeteOlcott wrote: Wed Aug 14, 2019 6:02 pm The best that I can tell from your replay is that the Ruby program proves that I am correct.
It demonstrably proves the exact opposite. It proves the law of non-contradiction as utter bullshit.
I have given you a system so powerful that it evaluates: P ∧ ¬P ⇔ ⊤

It's not magic. It's just levering the concept of a side effect.
You can't do that with pure functions.

Obviously, you are still welcome to choose consistency over expressive power. I can't tell you what to do...
Me? I am want my languages (formal or otherwise) to allow me to express myself concisely and readily.

Perfect is the enemy of good-enough. The beauty of para-consistent logics is that they don't explode.
Last edited by Skepdick on Wed Aug 14, 2019 6:28 pm, edited 1 time in total.
PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Truth can be understood as math

Skepdick wrote: Wed Aug 14, 2019 6:07 pm
Obviously, you are still welcome to choose consistency over expressive power.
Which is really only choosing not to be a Liar.
Skepdick
Posts: 4760
Joined: Fri Jun 14, 2019 11:16 am

### Re: Truth can be understood as math

PeteOlcott wrote: Wed Aug 14, 2019 6:28 pm Which is really only choosing not to be a Liar.
Bullshit. It's choosing a logic that doesn't explode.

Perfect is the enemy of good enough.
PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Truth can be understood as math

Skepdick wrote: Wed Aug 14, 2019 6:29 pm
PeteOlcott wrote: Wed Aug 14, 2019 6:28 pm Which is really only choosing not to be a Liar.
Bullshit. It's choosing a logic that doesn't explode.

Perfect is the enemy of good enough.
https://en.wikipedia.org/wiki/Principle_of_explosion
Logic that DOES explode incorrectly decides membership in the set of conceptual knowledge.
Skepdick
Posts: 4760
Joined: Fri Jun 14, 2019 11:16 am

### Re: Truth can be understood as math

PeteOlcott wrote: Wed Aug 14, 2019 6:54 pm https://en.wikipedia.org/wiki/Principle_of_explosion
Logic that DOES explode incorrectly decides membership in the set of conceptual knowledge.
Type theory explodes.

There is a very good reason you are speaking to me in English right now (which is para-consistent), and not in a formal language.

Because English is more expressive thanks to its inconsistencies.
PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Truth can be understood as math

Skepdick wrote: Wed Aug 14, 2019 7:00 pm
PeteOlcott wrote: Wed Aug 14, 2019 6:54 pm https://en.wikipedia.org/wiki/Principle_of_explosion
Logic that DOES explode incorrectly decides membership in the set of conceptual knowledge.
Type theory explodes.
You have not shown this with a concrete example of minimal complexity.
With the convoluted mess of the original incompleteness theorem proof it
is very easy for most everyone to get confused into believing that math is
incomplete.

So that you do not dismiss this as simplistic I quoted Wittgenstein's equivalent analysis:
When we realize that the whole Incompleteness proof can be boiled down to this self-contradictory
sentence: "This sentence is unprovable" then we realize the incompleteness is nothing more than a misconception.

Wittgenstein's equivalent analysis: (one of the founders of logical positivism)
I imagine someone asking my advice; he says: “I have constructed a proposition (I will use ‘P’ to designate it) in Russell’s symbolism, and by means of certain definitions and transformations it can be so interpreted that it says ‘P is not provable in Russell’s system’. Must I not say that this proposition on the one hand is true, and on the other hand is unprovable? For suppose it were false; then it is true that it is provable. And that surely cannot be! And if it is proved, then it is proved that is not provable. Thus it can only be true, but unprovable.”

Just as we ask, “‘Provable’ in what system?”, so we must also ask, “‘true’ in what system?” ‘True in Russell’s system’ means, as was said: proved in Russell’s system; and ‘false in Russell’s system’ means: the opposite has been proved in Russell’s system. – Now what does your “suppose it is false” mean? In the Russell sense it means ‘suppose the opposite is proved in Russell’s system’; if that is your assumption you will now presumably give up the interpretation that it is unprovable. And by ‘this interpretation’ I understand the translation into this English sentence. – If you assume that the proposition is provable in Russell’s system, that means it is true in the Russell sense, and the interpretation “P is not provable” again has to be given up.[…]
Skepdick
Posts: 4760
Joined: Fri Jun 14, 2019 11:16 am

### Re: Truth can be understood as math

PeteOlcott wrote: Wed Aug 14, 2019 7:22 pm You have not shown this with a concrete example of minimal complexity.
I gave you a paper and an implementation in OCaml.

Here is another implementation in Coq: https://github.com/UniMath/UniMath/blob ... sParadox.v

What more do you want?
PeteOlcott wrote: Wed Aug 14, 2019 7:22 pm Just as we ask, “‘Provable’ in what system?”, so we must also ask, “‘true’ in what system?”
You don't need to ask this question. You have admitted to using Type theory.

That's System F.

System F, also known as the (Girard–Reynolds) polymorphic lambda calculus or the second-order lambda calculus, is a typed lambda calculus that differs from the simply typed lambda calculus by the introduction of a mechanism of universal quantification over types. System F thus formalizes the notion of parametric polymorphism in programming languages, and forms a theoretical basis for languages such as Haskell and ML. System F was discovered independently by logician Jean-Yves Girard

That's why it's called Girard's paradox....
PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Truth can be understood as math

Skepdick wrote: Wed Aug 14, 2019 7:28 pm System F, also known as the (Girard–Reynolds) polymorphic lambda calculus or the second-order lambda calculus, is a typed lambda calculus that differs from the simply typed lambda calculus by the introduction of a mechanism of universal quantification over types. System F thus formalizes the notion of parametric polymorphism in programming languages, and forms a theoretical basis for languages such as Haskell and ML. System F was discovered independently by logician Jean-Yves Girard

That's why it's called Girard's paradox....
Great but you have not boiled that down to its minimal complexity.
Unless you can provide a complete concrete example of Girard's paradox
in a single simple English sentence it is not simple enough to be correctly
evaluated.

I am a C++ software engineer so I can probably get up to speed on
parametric polymorphism in a couple of minutes.
https://catonmat.net/cpp-polymorphism Done !
Skepdick
Posts: 4760
Joined: Fri Jun 14, 2019 11:16 am

### Re: Truth can be understood as math

PeteOlcott wrote: Wed Aug 14, 2019 8:07 pm Great but you have not boiled that down to its minimal complexity.
Unless you can provide a complete concrete example of Girard's paradox
in a single simple English sentence it is not simple enough to be correctly
evaluated.
As always - show me your evaluator and I will break it for you.
PeteOlcott wrote: Wed Aug 14, 2019 8:07 pm I am a C++ software engineer so I can probably get up to speed on
parametric polymorphism in a couple of minutes.
https://catonmat.net/cpp-polymorphism Done !
And I am a software engineer who is language-agnostic. C++ does not have parametric polymorphism. It has a templating hack which happens at compile time. Not at runtime.