Can somone redpill me on the link between language and mathematics?

http://vixra.org/abs/1506.0055

>>9054184
>http://www.helsinki.fi/esslli/courses/readers/K54.pdf

File: 9781107606753.jpg (33KB, 372x400px) Image search: [Google]

33KB, 372x400px

>>9054184
First-order logic is the language of mathematics.
The easiest way to swallow the redpill (that I know of) is probably pic related.
Peter Smith seems to be quite good at bringing these kinds of ideas to non-specialists, in my experience (you can check out his blog "Logic Matters" for prepatory material if you need help bridging the gap).

File: 714px-Chomsky-hierarchy.svg.png (40KB, 714x515px) Image search: [Google]

40KB, 714x515px

>>9054194
>First-order logic is the language of mathematics.
>First-order logic is a language
kek

>>9054202
>>First-order logic is a language
It is tho...

>>9054233
It really depends on your definition of language.

>>9054235
https://en.wikipedia.org/wiki/Formal_language

>>9054243
Yeah, that's a loose definition of 'language'. This is not what the term means to linguists. Otherwise, thereù's no distinguishing French from C++, which are very different types of formal systems.

>>9054248
Not that guy but I see more similarities than differences between them desu
There are well-defined standards for both C++ and French (and mathematics) which states what kinds of terms are grammatically valid, which can be used to resolve any disputes that arise.
And these disputes arise in practice because digesting the whole standard and consistently assuring 100% compliance is a pain, so plenty of people do write implementation-specific C++ / speak informal French (and write non-rigorous mathematical proofs).

Semantically, programming languages have "abstract syntax trees", first-order logic has "terms", and natural languages have "parse trees" (or whatever linguists who study phrase structure grammar call it, I'm not formally trained in linguistics).
In fact you could probably make similar claims for the presence of syntactic analogies as well, except that the semiotic mapping from syntax to semantics is clearer for programming languages and mathematical logic, and less so for natural languages.

>>9054261
Yes, they are both formal systems which can be defined recursively. But a child exposed to spoken C++ will not learn to speak C++, while a child exposed to French stimuli will learn French. Languages as formal systems are assumed to rely on a universal foundation upon which they can be derived. This universal foundation is what we call Language, which is distinct from the foundations of say C++, Java or Python. No child is ever going to speak Python.

>>9054261
>>9054271
To elaborate further, the starting point in determining what a language is is to define it precisely as one of those systems children learn and use to communicate. What is left is then to determine what the formal characteristics of such a system are.

>>9054277
>the starting point in determining what a language is is to define it precisely as one of those systems children learn and use to communicate
For natural languages, sure. But for programming languages that definition won't work because there is no analogue of "children" in that context.
We have "computers" though, and if you equate "learning" with "programming" (say, writing a compiler) then the syntactic substitution of "children" for "computer" in the definition seems to work pretty well.

The word "language" is overloaded with linguists using one definition (>>9054248) and mathematicians/programmers using another. I'm not saying that they're completely exchangeable, but there are enough similarities to draw "useful" analogies between the two contexts. I use the word "useful" in the sense that people studying languages (in one sense of the word) can benefit from reading research on language (in the other sense) for insights -- that's pretty much my position as a mathfag with no training in linguistics.
For this reason I personally support the ambiguity in the definition of "language" since it serves a pragmatic purpose in exposing me to new and potentially useful avenues of research -- this is the informal layperson usage of pragmatism, not the linguistic definition. I can understand if others have differing preferences though.