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I'm trying to explain the limits of a function in calculus and I'm not sure whether to mention that the notation that to state that 'the limit of f(x) as x approaches zero' is a slight use-mention distinction example as we use x as a possible denotation of a number, and then later as a 'variable' a symbol which approaches a particular value.

Are we using 'x' differently (one as an unassigned object) the other as a symbol which may take assignments (the behaviour under these assignments being the object of study).

Confused
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  • Someone once asked here if every use of a symbol such as '2' is actually distinct. As I read your question, I think "this variable" in place of "x". Maybe that helps? – Scott Rowe Dec 06 '22 at 16:23
  • @Confused See https://philosophy.stackexchange.com/q/94749/33265 on referring to mathematical concepts. – Speakpigeon Dec 06 '22 at 16:39
  • @Speakpigeon In your answer on this thread you say that 'x' denotes a 'the idea of a mathematical variable', I was confused by this, I would say an unassigned variable denotes nothing but *represents* a mathematical variable. What did you mean by that? – Confused Dec 06 '22 at 19:45
  • No, it is an example of standard ambiguity resolved by context. And it is already in the functional notation, even without limits: *f(x)* can mean either the map *f* with variable *x* or the image of *x* under *f* depending on context, see [Why does notation for functions seem to be abused and ambiguous? on Math SE](https://math.stackexchange.com/q/1102156/152568). More generally, symbols in mathematics are used ambiguously, as uninterpreted symbols of formal theory or as labels of their [interpretations on a model](https://en.wikipedia.org/wiki/Model_theory#Basic_model-theoretic_concepts). – Conifold Dec 07 '22 at 03:32
  • @Conifold True, this ambiguity in the function notation is one thing I do point out. I would say, perhaps it is an example where the difference between 'use' and 'mention' is something that can be shown by context, it does not make much sense to talk about a number approaching anything. – Confused Dec 07 '22 at 09:42
  • @Confused "*What did you mean by that?*" The word "apple" is used to denote apples, which are objects in the world. Mathematical terms do not denote objects in the world but mathematical concepts, i.e., ideas in our mind. The figure '2' denotes the number 2, which is an abstraction, i.e., an idea in our mind. The term 'x' denotes the concept of mathematical variable, which is an idea in our mind. – Speakpigeon Dec 07 '22 at 16:57
  • @Speakpigeon my definiton of 'denote' is wrong, perhaps I mean 'refer' does 'x' (name) any particular object in your view? If not, then definitely it's just my understanding of the meaning of 'denote' is wrong. – Confused Dec 07 '22 at 19:02
  • For example 'John' denotes a man for me if there is one man named John I am referring to, however 'x' doesn't name any object. I apologise if this is just my own lack of english. – Confused Dec 07 '22 at 19:05
  • I suppose that you can see it this way if you think of uninterpreted symbols as "mentions" of the objects they interpret. However, in formal theories uninterpreted symbols are used in their own right with no interpretation attached (the point is that they can, in fact, have multiple interpretations), so I do not think that the use/mention distinction is particularly suitable in this context. "Number approaching" may not be meaningful, but "*x* approaching" can be used formally, with *x* just a symbol, if one sets up formal rules for manipulating limits, for example. – Conifold Dec 08 '22 at 08:42
  • @Conifold I actually think we're making the same point, we 'mention 'x' a symbol, or element in our language, in f(x) we interpret it as a 'number'(using), but the 'x approches' it only makes sense if we use x just as a symbol (mentioning), that's why I say there is a distinction, in one case we talk about a function on a number which we call x, on another we talk about a symbol (variable). As you correctly point out, this is all contextual, but in formal languages we would probably have to switch to meta-language to discuss variables 'approaching' anything. – Confused Dec 08 '22 at 11:00
  • @Confused In mathematics, 'x' is used to denote (name, refer etc.) the mathematical concept of variable. Concepts are not objects, so 'x' does not denote (name, refer etc.) any object. Which is why learning mathematics is a bit more difficult than learning English. – Speakpigeon Dec 08 '22 at 11:03
  • @Speakpigeon in terms of denoting or naming, this is where I get confused, so '1' denotes or names a number, but 'x' denotes a variable, yet 1+x is not the addition of a number and a variable? as '+' is defined only on numbers. – Confused Dec 08 '22 at 11:16
  • @Speakpigeon but then again, I might be confused because I assume to 'denote' is to name at object level, whereas in language it might server other purposes like to write [insert here] in a sentence denotes that without [insert here] being an object. – Confused Dec 08 '22 at 11:23
  • @Confused "*yet 1+x is not the addition of a number and a variable*" Indeed, it is not and nobody knows the result of 1 + x. 1 + 2 denotes the addition of the number denoted by the figure '1' with the number denoted by the figure '2'. 1 + x denotes the addition of the number denoted by the figure '1' with the number which is the value of the variable denoted by the name 'x'. What is the value of the variable denoted by 'x'? We don't know of course but if we did, we could do the maths. – Speakpigeon Dec 08 '22 at 17:07
  • @Speakpigeon so it is possible to denote a mathematical variable without having it as one of the things at 'object' level (one of the numbers, sets etc actually *being discussed*) – Confused Dec 10 '22 at 12:34
  • @Speakpigeon I see how that can be a possibility, so in your view the variable exists purely at language level, but still can be denoted? – Confused Dec 10 '22 at 15:07
  • @Confused This is not purely at language level. A variable x is not a word. It is a concept (i.e. in our mind) denoted by the word 'x'. We need language to talk about it like we need language to talk about anything from cats to dogs to our mental states. – Speakpigeon Dec 10 '22 at 16:27
  • @Speakpigeon however, x denotes a variable but does not denote any mathematical object in our set of discussion such as 1,2,3 or the sine function, it is of a higher level and part of our discussion of the universe, not part of the universe? What I'm asking is can we denote a variable without denoting any object in our universe, as when we use an unassigned variable it does not name any number. – Confused Dec 11 '22 at 09:49

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