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by PeteOlcott
Fri Sep 13, 2019 11:59 pm
Forum: Logic and Philosophy of Mathematics
Topic: Who knows lambda calculus?
Replies: 37
Views: 572

Re: Who knows lambda calculus?

Anyone striving to derive strong AI must formalize human inference and knowledge correctly. Why? If you build strong AI it can acquire its own knowledge-base. Like babies do. If math does this incorrectly then math much be fixed. You can't define 'correctness' without ending talking about morality....
by PeteOlcott
Fri Sep 13, 2019 11:51 pm
Forum: Logic and Philosophy of Mathematics
Topic: Who knows lambda calculus?
Replies: 37
Views: 572

Re: Who knows lambda calculus?

None of the has anything to do with incompleteness. Who cares about theoretical completeness except theoreticians? In the practical world your knowledge is always incomplete. There is no proof you can offer to convince me otherwise. Semantic truth is more important than syntactic truth. Beware of b...
by PeteOlcott
Fri Sep 13, 2019 11:35 pm
Forum: Logic and Philosophy of Mathematics
Topic: Who knows lambda calculus?
Replies: 37
Views: 572

Re: Who knows lambda calculus?

None of that has anything to do with sound deductive inference. Sound deductive inference requires truth preserving operations to be applied to true premises deriving truth. Any formal system diverging from this model is incorrect. So non-deterministic/probabilistic algorithms are "incorrect"? Idio...
by PeteOlcott
Fri Sep 13, 2019 11:07 pm
Forum: Logic and Philosophy of Mathematics
Topic: Who knows lambda calculus?
Replies: 37
Views: 572

Re: Who knows lambda calculus?

None-the-less it proves that formal systems are not incomplete or inconsistent, math has no entropy. It proves no such thing. Mathematicians are entropy to maths. Interpretation is Operational Semantics. And denotationally, the entire field of stochastics/probability theory is about entropy. None o...
by PeteOlcott
Fri Sep 13, 2019 11:00 pm
Forum: Logic and Philosophy of Mathematics
Topic: Who knows lambda calculus?
Replies: 37
Views: 572

Re: Who knows lambda calculus?

A sequence of sound inference beginning with true premises derives true conclusions. That is not true in a non-deterministic universe. Deductive reasoning can lead to false conclusions with entropy. None-the-less it proves that formal systems are not incomplete or inconsistent, math has no entropy.
by PeteOlcott
Fri Sep 13, 2019 10:46 pm
Forum: Logic and Philosophy of Mathematics
Topic: Who knows lambda calculus?
Replies: 37
Views: 572

Re: Who knows lambda calculus?

Everyone understands that arithmetic of natural numbers concurrently defines true and provable. We only know that 2 + 3 = 5 because the axioms prove this. I can trivially misunderstand this for the sake of contrarianism. But I am not in the mood. Why do you want to blur the lines between truth and ...
by PeteOlcott
Fri Sep 13, 2019 10:09 pm
Forum: Logic and Philosophy of Mathematics
Topic: Who knows lambda calculus?
Replies: 37
Views: 572

Re: Who knows lambda calculus?

To define the "=" relation between finite strings of numeric digits for the +(S1, S2) function requires an algorithm. Why do you want to define the "=" relation? There are plenty of algorithms possible where "=" is not required. Everyone understands that arithmetic of natural numbers concurrently d...
by PeteOlcott
Fri Sep 13, 2019 8:16 pm
Forum: Logic and Philosophy of Mathematics
Topic: Who knows lambda calculus?
Replies: 37
Views: 572

Re: Who knows lambda calculus?

Reflection is not required By whom and for what purpose? I USE reflection when I construct formal systems (a.k.a program) every damn day. As far as I am concerned if it can't DO reflection it's not a formal system. In fact - it's such a strong criterion that as far as I am concerned Mathematics is ...
by PeteOlcott
Fri Sep 13, 2019 7:45 pm
Forum: Logic and Philosophy of Mathematics
Topic: Who knows lambda calculus?
Replies: 37
Views: 572

Re: Who knows lambda calculus?

It seems to me that HOL can express any true thing, and HOL can be expressed as a type-2 language. Type-2 languages cannot do reflection But you are really fucking confused about what it is that you are trying to do. Lambda calculus is the formalisation of a Turing machine. Lambda calculus can RECO...
by PeteOlcott
Fri Sep 13, 2019 7:42 pm
Forum: Logic and Philosophy of Mathematics
Topic: Who knows lambda calculus?
Replies: 37
Views: 572

Re: Who knows lambda calculus?

Skepdick wrote:
Fri Sep 13, 2019 7:22 pm
PeteOlcott wrote:
Fri Sep 13, 2019 7:19 pm
(2) What true thing about integers cannot be expressed in a type-2 language?
A declarative query which answers the question: "How many facts does this formal system know?"
Some of these facts are algorithmically compressed as algorithms, thus the list is finite and countable.
by PeteOlcott
Fri Sep 13, 2019 7:40 pm
Forum: Logic and Philosophy of Mathematics
Topic: Who knows lambda calculus?
Replies: 37
Views: 572

Re: Who knows lambda calculus?

Skepdick wrote:
Fri Sep 13, 2019 7:22 pm
PeteOlcott wrote:
Fri Sep 13, 2019 7:19 pm
(1) To express all of the functionality of any formal system.
How do you express Turing-completeness in a Type-2 grammar?
I don't see any case where the conventional notion of a formal system requires Turing completeness.
by PeteOlcott
Fri Sep 13, 2019 7:19 pm
Forum: Logic and Philosophy of Mathematics
Topic: Who knows lambda calculus?
Replies: 37
Views: 572

Re: Who knows lambda calculus?

The notion of a formal system may not require a type-0 language a type-2 language may be sufficiently expressive. That is not a complete sentence. Sufficiently expressive to...... do what ? Obviously a type-2 language is NOT sufficiently expressive to say true things about the integers. (1) To expr...
by PeteOlcott
Fri Sep 13, 2019 7:08 pm
Forum: Logic and Philosophy of Mathematics
Topic: Who knows lambda calculus?
Replies: 37
Views: 572

Re: Who knows lambda calculus?

The point was to define at the most abstract level the syntax and semantics the most generic notion of a formal system from which every possible formal system could be defined. https://en.wikipedia.org/wiki/Chomsky_hierarchy#Type-0_grammars Type-0 grammars include all formal grammars. You know what...
by PeteOlcott
Fri Sep 13, 2019 6:42 pm
Forum: Logic and Philosophy of Mathematics
Topic: Who knows lambda calculus?
Replies: 37
Views: 572

Re: Who knows lambda calculus?

We wasted all this time over trivialities. The point was to define at the most abstract level the syntax and semantics the most generic notion of a formal system from which every possible formal system could be defined. We can coalesce the notions of functions and predicates into a function that may...
by PeteOlcott
Fri Sep 13, 2019 5:22 pm
Forum: Logic and Philosophy of Mathematics
Topic: Who knows lambda calculus?
Replies: 37
Views: 572

Re: Who knows lambda calculus?

You lost track of the distinction between syntax and semantics. I didn't lose track. You don't understand the distinction. Now you are a liar, going back and changing what you said and claiming that you never said it! You are trying to define the syntax of Lambda calculus (Type 0 grammar) using BNF...