User blog comment:KthulhuHimself/Library Function/@comment-2033667-20151104065208

RE: "If FOST is mathematics, this is mathematics. Both are logic-based languages."

Here is part of the definition of FOST:

∀R { {    ∀[ψ], s: R([ψ],t) ↔ ([ψ] = "xᵢ ∈ xⱼ" ∧ t(xᵢ) ∈ t(xⱼ)) ∨ ([ψ] = "xᵢ = xⱼ" ∧ t(xᵢ) = t(xⱼ)) ∨ ([ψ] = "(¬θ)"   ∧ ¬R([θ], t)) ∨ ([ψ] = "([θ]∧ξ)" ∧ R([θ], t) ∧ R([ξ], t)) ∨ ([ψ] = "∃xᵢ(θ)" ∧ ∃t′: R([θ], t′)) (where t′ is a copy of t with xᵢ changed) } ⇒ R([ϕ],s) }

Not all logic needs to look like this. But I see no logic in your blog post, or at least not in terms I understand with my mathematician's hat on.

Don't feel obliged answer my last comment. I realize now it was a bad idea to start out asking questions on every term I don't understand; I'll just frustrate us both. Instead I will try to deconstruct the original post as thoroughly as possible and explain best practices for communicating mathematical ideas.

Order of definitions: It is always a good idea to introduce definitions incrementally. If you want to define A, and the definition of A relies on B, define B first and then define A. Reference past definitions. If you need to reference future definitions, chances are you need to reorder your writing. There are probably exceptions to this, but do the best you can to work in chronological order. In this case, you should reverse the introduction of the definer language and Lib(n). The definition of your "target function" does not need to come first, and in general it shouldn't. The same applies to the "Definer language" section, where you use the term "formal English" before you define it. Swap 'em. It will be much clearer that way.

Formal English: Referring to a dictionary in the context of formal mathematics poses several problems. A dictionary is inherently circular. It defines English words in English. How do you interpret the definitions of the words? You use formal English, which is defined by the Oxford Advanced Learners' Dictionary. How do you interpret the definitions of the words in that dictionary? You use formal English, which is defined by the...

Dictionaries do not, in general, describe grammars or formal linguistic properties of languages. They are meant to be human-readable references describing how to use individual words and phrases. A dictionary might tell you what each word individually means, and it may give us some tips on properly combining its entries, but it doesn't teach us how to use the English language in general. Can you determine from only a dictionary that "bort slap clam mouth" is not a grammatically correct sentence?

Any formal form of English would have to do away with the vast majority of terms we use in our daily lives. How do you define a tree (the plant) in an abstract universe? I guess you could select some mathematical model of quantum physics and attempt to formalize atoms and molecules and DNA. But formally, mathematically defining a tree? Good luck with that. The vast majority of the terms we use to define physical objects are vague compared to the extremely clear-cut and precise formulations we see in mathematics.

Definer language: What do you define as an English letter? The word "naïveté" is English; are ï and é English letters? Are uppercase and lowercase letters allowed?

"any other symbol whatsoever, as long as it is defined in 'definer' somewhere else in the expression." Whether something is "defined" is incredibly subjective and you need to be explicit about what this means.

Lib(n): What's a number? Are complex numbers numbers?

And here is the #1 most important point: the critical parts of this definition — "well-defined" and "expressed by" — are subjective and unexplained.

I was going to elaborate more on those last two sections but I need to get to bed early. Peace.