What are the limitations of the language of mathematics?

Question

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.

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 03:14
Jokes, puns, poetry come to mind (as language-able but not mathematizable) — Rushi, Aug 16 '20 at 07:16
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
@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

Speakpigeon · Answer 1 · 2020-08-16T09:50:08.750

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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.

edited Aug 16 '20 at 09:50

answered Aug 16 '20 at 09:39

Speakpigeon

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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 09:54
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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" – 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
@ask_hole I clarified. You are just not interested. – Speakpigeon Aug 17 '20 at 11:06
why do you think it is the "only" difference? "The fundamental differences between natural language and mathematical language derive **first and foremost** from the fact that mathematical language is more precise" why do you think any of what you claim? it would be as foolish to attempt to write a love poem in the language of mathematics as to prove the Fundamental Theorem of Algebra using the English language." – Aug 18 '20 at 09:58
i'm also baffled what makes you statement a "joke", rather than something you wrote on the internet? i mean here is a poem i wrote 2+1=3 great – Aug 18 '20 at 10:07
i'm unconvinced it's a funny joke... i suppose if we are expecting a difference between material and logical implication, then seeing it negated might amuse. i don't think it suffices for a funny joke tbh and i think you need a longer set up, one that suggests a lack of logical mplication – Aug 18 '20 at 11:11
even with a longer set up, i'm unconvinced it would be funny without the clowning. (φ → ψ) is a vacuous inference, and it's unclear to me that can be a punch line – Aug 18 '20 at 11:49
@Speakpigeon - If you're using a notation where the arrow stands for logical implication, then this would translate to something like "(φ materially implies ψ) logically implies (φ logically implies ψ)", no? And isn't that statement just false? There are plenty of material implications that are specific to a particular domain of discourse (say, describing traits of a particular group of people in a room), whereas a logical implication should be true in all possible worlds where the premise is true. – Hypnosifl Aug 19 '20 at 20:07
@Hypnosifl I am not sure I understand your point, but maybe you could provide examples? Given what you say, this should be easy for you. – Speakpigeon Aug 20 '20 at 10:54
Suppose the domain of discourse is a collection of people in a room as I said, and it happens to be true of this group of people that all the people with brown hair are wearing sneakers. In this case, the material implication "For all x, brownhair(x) -> wearingsneakers(x)" is true, and likewise if "a" represents a particular brown-haired person, brownhair(a) -> wearingsneakers(a) is true. But this does not mean that the property of having brown hair *logically implies* wearing sneakers, one can't deduce the latter from the former regardless of the context (the specific domain of discourse). – Hypnosifl Aug 20 '20 at 14:07
@Hypnosifl 1. What you say here is that A → B doesn't imply C → D. And I agree with you. - 2. However, this has nothing to do with the material implication and/or with the logical implication. 3. Why are you talking of the material implication in the first case and of the logical implication in the second? Maybe you think that the logical implication doesn't apply to particular situations? Yet, given your example, it is true that if x is in the room in question, then if x has brown hair, then x is wearing sneakers. And this is therefore a true logical implication. – Speakpigeon Aug 20 '20 at 16:50
@Hypnosifl We can say the same thing differently: "*x is in the room in question*" logically implies ("*x has brown hair*" logically implies "*x is wearing sneakers*"). Yes? No? – Speakpigeon Aug 20 '20 at 16:53
Logical implication depends only on the logical form of the premises, not the domain of discourse (of course you can explicitly make statements about the domain of discourse into premises, but there is no requirement that you do so for material implication). Another way of stating this, as mentioned on p. 5 at https://www2.seas.gwu.edu/~ayoussef/cs1311/Logic.pdf is that p logically implies q is equiv. to the statement that p->q is a tautology (where -> stands for material implication). brownhair(a) -> wearingsneakers(a) is not a tautology, so the first doesn't logically imply the second. – Hypnosifl Aug 20 '20 at 17:59
If F then U. If U then C. If C then K. Assume F. Therefore, by transitivity, FUCK. (Feel like I have seen this before but don't recall where.) – Kristian Berry Sep 07 '20 at 01:23
@KristianBerry This is the kind of joke most mathematicians probably understand. Mine is a little bit more challenging. – Speakpigeon Sep 07 '20 at 10:02
https://www.lesswrong.com/posts/j42LsXe3qkHhctezz/brain-in-a-vat-trolley-question-0 has my favorite philosophy joke, which is also how I learned that I like philosophy (first time I read it, I didn't recognize all the Easter eggs, but I saw the style and was like, holy jokes Batman, that's the style of my thoughts). – Kristian Berry Sep 07 '20 at 13:04

score 1 · Answer 2 · answered Aug 16 '20 at 20:13

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.

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. — , Aug 17 '20 at 03:41
@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

score 0 · Answer 3 · answered Aug 16 '20 at 13:10

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.

score 0 · Answer 4 · answered Aug 16 '20 at 20:33

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.

What are the limitations of the language of mathematics?

4 Answers4