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I was told that mathematics cannot express qualitatively what the elements of a set are, such that you cannot say for example that the members of a set consists of white tigers. So mathematics cannot add qualitative details to a mathematics concept or a mathematics instance. I would like to know what are some of the other limitations of the language of mathematics compared to written or spoken language such as English.

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    Mathematics (typically) does not care what the elements are, but it can express it in the same way that the natural language does, by adding a predicate "white tiger" and asserting it of the elements. If you think about it, natural language by itself does not express anything either, it only strings together labels. It is the connection of labels to actions that does the expressing, and one can connect them to mathematics just as well. Natural languages just have a longer pre-labeled vocabulary. – Conifold Aug 16 '20 at 3:14
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    Jokes, puns, poetry come to mind (as language-able but not mathematizable) – Rusi-packing-up Aug 16 '20 at 7:16
  • I wonder if jokes and irony enter into the Turing test. Can Deep Blue make up new jokes or know when to laugh or recognize irony? – Nelson Alexander Aug 16 '20 at 20:19
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    @NelsonAlexander The Turing Test is about the ability to respond to arbitrary statements, so yes, the questioner could request a joke. Deep Blue is a chess engine, so I suppose it depends on what you consider to be a joke. "Why did the knight move to D4? Because ND4 = \argmax_{move} minimax(move, position_t)!?" – Ray Aug 18 '20 at 16:53
  • Ha! Well, requesting "a joke" would be pretty easily tagged, but recognizing one might be hard. I didn't know Deep Blue was chess only, I thought it also did restaurant reservations. – Nelson Alexander Aug 18 '20 at 20:15
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The mathematical language is simply a more rigorous way to talk about the world. There is no limitation to it in this respect that wouldn't be a limitation to any language.

That nobody knows today how to express jokes, puns and poetry mathematically does not imply that they could not possibly be expressed mathematically. There was a time when nobody knew how to express probabilities mathematically, for example, and look now...

The fact that there are no poems written in the mathematical language does not imply that this could not be done. Rather, it seems a direct consequence of the fact that it is a specialised language and that therefore most people don't understand it well enough.

As to jokes, here is one, written in the language of formal logic:

(φ ⊃ ψ) → (φ → ψ)

It is actually very funny, but you need to understand it and very few people get it.

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    The joke is just that you're using two different notations for material implication? I wouldn't really say that's a joke expressed "in the language of formal logic" because there isn't any real formal logic system that uses both notations at once, it's more like a history-of-logic joke. – Hypnosifl Aug 16 '20 at 9:54
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    “Due to the recent A-level fiasco in the UK, every applicant was mistakenly initially rejected by Cardiff and Vale University College. Students were appalled to learn about this Uni-Vale-n’t-s axiom.” – Sofie Selnes Aug 16 '20 at 11:37
  • the fact you're not googling before typing your thinking makes it no better "Thus a 'poem' might 'borrow' the language of mathematics, using numbers, alone or mixed with words. (Willard Bohn gives a 'numbers only' example by Picabia in his introduction to The Dada Market, an anthology" – user47711 Aug 16 '20 at 15:07
  • @ask_hole I have to wonder why you could not provide any example of an actual mathematical poem. You know, real mathematics? Mathematics that actually says something mathematical. I myself gave one example of a joke, but it is a unique example, for the occasion. – Speakpigeon Aug 16 '20 at 17:47
  • @Hypnosifl No, not two different notation for the same thing. In (φ ⊃ ψ) → (φ → ψ), the horseshoe '⊃' as usual stands for the material implication, while the arrow '→' as usual stands for the logical implication.And it is expressed in the language of formal logic, just not one you would know. As I said, few people get it. – Speakpigeon Aug 16 '20 at 17:52
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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.

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    Besides heartily supporting Nelson piercingly apt depiction of the abyss which gapes between the beauty, richness and fullness of human language and the purely secondary, descriptive nature of mathematics, it might be added that without people and objects in nature there would be no mathematics as there would be nothing to measure or compute. As Spinoza noted, mathematics is merely an aid to the imagination. – user37981 Aug 17 '20 at 3:41
  • @NelsonAlexander "Mathematics attempts to void itself of as much experiential content as possible" Yet, mathematics is fundamentally rooted in the cognitive processes of the human mind. There is no mathematics which is not an expression of some logical relation, and logic is nothing if not the logic of the human mind. – Speakpigeon Aug 17 '20 at 16:25
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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.

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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.

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