수학 언어의 한계는 무엇입니까?

Question

나는 수학이 세트의 요소가 무엇인지를 질적으로 표현할 수 없다고 들었는데, 예를 들어 세트의 구성원이 백호로 구성되어 있다고 말할 수는 없습니다. 따라서 수학은 수학 개념이나 수학 사례에 질적 세부 사항을 추가 할 수 없습니다. 나는 영어와 같은 서면 또는 구어와 비교하여 수학 언어의 다른 한계가 무엇인지 알고 싶습니다.

Answer 1

수학적 언어는 단순히 세상에 대해 이야기하는보다 엄격한 방법입니다. 이 점에서 어떤 언어에도 제한이없는 제한이 없습니다.

오늘날 농담, 말장난,시를 수학적으로 표현하는 방법을 아무도 모른다고해서 수학적으로 표현할 수 없다는 의미는 아닙니다. 예를 들어, 확률을 수학적으로 표현하는 방법을 아무도 몰랐던 때가있었습니다.

수학적 언어로 작성된 시가 없다는 사실이 이것이 불가능하다는 것을 의미하지는 않습니다. 오히려 그것은 전문화 된 언어이기 때문에 대부분의 사람들이 그것을 충분히 잘 이해하지 못한다는 사실의 직접적인 결과로 보입니다.

농담에 관해서는 공식 논리 언어로 작성된 농담이 있습니다.

(φ ⊃ ψ) → (φ → ψ)

실제로는 매우 재밌지 만 이해해야하며 이해하는 사람은 거의 없습니다.

Answer 2

Contrary to some commenters here, there is a vast difference between mathematics and language, despite the fact that any sentence can obviously be translated into mathematized "information."

Russell, the Logical Positivists, and others set out to rid language of its murky qualities by reducing both language and mathematics to logic. While the work was quite fruitful, the project itself was deemed a failure, at least as a complete system. The break between early and late Wittgenstein offers a dramatic encapsulation of this "failure," given the vast, complex, living, and performative nature of language.

In the first place, language is embodied, experiential, and primarily oral. It begins with vibrations in the womb and is continuous with human life, physical contexts, and reproduction. We can transcribe words into visual alphabets, but these require a rather unnatural, arduous process of learning. You cannot translate these visual signs back into language without access to the spoken words. Apart from crude pictograms, you cannot translate or recover a "dead language" such as Linear A without some relation, however indirect, to a living "spoken" language.

This suggests that language has the same sort of time-bound irreversibility as life itself, whereas mathematics is "reversible" and hence empty of meaning, if "meaning" has to do, as Luhmann says, with relations of actual to possible. Mathematics attempts to void itself of as much experiential content as possible, whereas language is experience and always assumes, however remotely, an embodied speaker with a particular history and environment.

We cannot learn mathematics without language, but we readily learn language without mathematics. In theory, of course, some might argue that AI would entail a mathematization of the unique human language skills that move within and between brains. But one of the linguistic capacities of intelligent brains is that they reproduce themselves, while it is very doubtful that computing machines can reproduce themselves outside of an environment of reproducing humans.

Answer 3

There is an important distinction between pure mathematics and applied mathematics.

Pure mathematics is concerned entirely with abstract truths of the general form "given certain initial formal conditions or postulates, what are the consequences?" For example in an axiomatic system these formal conditions are divided into primitives, relations, and axioms which define how the relations apply between primitives. But the primitives and relations have no intrinsic meaning.

When some meaning is applied to a primitive, the exercise becomes one of applied mathematics. A given pure mathematical discipline may be ascribed many different meanings, each leading to a different branch of applied mathematics. As David Hilbert once apocryphally remarked of axiomatic geometry, one might perfectly well apply "points", "lines" and "planes" to tables, chairs and beer mugs.

Thus the mathematical properties of the elements of a set, as primitive placeholders, is the domain of pure mathematics, while the mathematical properties of a cageful of white tigers is the domain of applied mathematics.

Answer 4

There's a lot of solid mathematics behind colors and music. In set theory, you can talk about sets with different transfinite cardinals for their number of colors.

Logical structure can be diagrammed, in general and for specific concepts.

Still, I would hedge my bets and just say that we don't know whether we can associate every relevant concept with its own mathematicization, in a relevant way. In cases where success does not seem forthcoming, it may be that we just haven't figured out the word problem yet, so to speak.