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I have been wondering about the formalism of what 'exactly' a variable is, and its role in proofs in mathematics. I have seen a few questions here, such as this one: Is there a way of defining the notion of a variable mathematically?. However, I have not fully absorbed the responses to those questions, and I believe this question regards somewhat different notions.

Wikipedia defines a variable as "In mathematics, a variable is a symbol used to represent an arbitrary element of a set". There was a great question (and answers') about the role of arbitrary elements in sets, The role of 'arbitrary' in proofs, and it seems to me that per the definition of a variable, it is exactly a symbol to represent some 'arbitrary' element.

My motivation for this question (and my confusion) comes from the role of using arbitrary elements in proofs. I understand that something of the form $P(x)$ is not a statement, it is an open sentence. However, something of the form $P(2)$ is a statement, since we know exactly the mathematical object we are dealing with. For example the statement, $\forall x \in X. P(x)$ means we have a collection of precise statements $$P(x_1) \land P(x_2) \land P(x_3)....$$

Where now the symbols $x_1, x_2, x_3, ...$ represent a precise and distinct mathematical object, for example the numbers $1,2,3,...$ if the set that we quantify over is $\mathbb{N}$.

The trouble in my mind however is we sometimes have proofs of the form, "Let $x$ be some element in $S$. Then... [reasoning about $x$] $\implies$ [some conclusion]. In this case, we are reasoning about $x$ as we would a particular mathematical object $x_1$, but I don't quite understand how we would do this?

To expand on my doubt: What I am confused by is that for a mathematical object (such as an integer) we have precise and formal definitions/constructions/axioms for these objects. Numbers can be defined in terms of sets, so can ordered pairs, etc. So when we reason about these objects, we are reasoning about some particular well defined 'mathematical object'. But with arbitrary elements/variables, it seems to me that we are saying "OK, this symbol refers to some object in this set, but you don't actually know which one". Since this is the case, how could we construct a definition of such an object, that wouldn't itself need to be expressly defined in terms of ALL the objects it could possibly be? And if indeed it is the case that we cannot, then how can we reason about this 'thing' as if it were a precisely defined mathematical object?

Or does this mean there is some definable set for example, which contains an element $x$ that is an 'arbitrary element' of $\mathbb{Z}$?

My apologies if this question is not a good fit for this stack exchange, I am not sure if this is more related to the philosophy of mathematics, but I have seen a few similar questions on the topic so I thought I would post it here.

EDIT: I had a look at the other question, and I do not believe mine is a duplicate (and I don't think the answers clarified my doubts, though I have perhaps not explained myself properly).

For example in document which the accepted answer quotes of, it says two interesting points: "Roughly speaking, a variable is a symbol for which one substitutes names for some objects, usually a number in algebra. A variable is always associated with a set of objects whose names can be substi- tuted for it. These objects are called values of the variable. (p. 70)" and on another page "There is a subtle quandary here. We want students to have the referents (usually real numbers) for variables in mind as they use the variables. But we also want students to be able to operate on the variables without always having to go to the level of the referent."

However, my doubt is not on what we are doing, or even why we are doing it, but on how we can manipulate this symbol as if it were a referent (as that document puts it). It seems to me a symbol that can be substituted for any elements in a set is different from being an element of said set, so why do we perform the same operations, and use the same logic, on both?

Mauro ALLEGRANZA
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masiewpao
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    Ultimately this is clarified by the notion of formal proof: basically, variables are purely "syntactic" objects and the way to understand them is to understand the rules of proof which govern how we manipulate them. There are many different (equivalent in a precise sense) approaches; I personally like sequent calculus. If you look at that and it seems potentially interesting, I can expand this comment into an answer. – Noah Schweber Oct 25 '19 at 02:42
  • @NoahSchweber I am sorry this is a very late reply, but I think I somehow missed this. I indeed would be very interested in such an answer. – masiewpao Dec 20 '20 at 13:05

1 Answers1

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what 'exactly' a variable is ?

See the post Why is a variable called variable in mathematics.

and [what is] its role in proofs in mathematics.

Variable are used in open formulas:

"... something of the form $P(x)$ is not a statement, it is an open [formula]."

The details of the semantics of the language dictate "how to read" a formula with free variables.

Usually, we use a "context" [technically called: variable assignment function], i.e. a way to assign a "temporary meaning" to the free variables.

We can compare a free variable to a pronoun of natural language.

To assert "$x$ is Prime" is the same as "it is Prime": its meaning depends on what the context assigns to "it".

In the same way, in roder to read formula $P(x)$, we have to specify an interpretation, e.g. the domain of natural numbers for the quantifiers and the property "to be Prime" for the predicate $P(x)$.

Having done this, the truth value of the formula $P(x)$ and will depend on the "object" we will assign to "it" as reference.

Proof systems for first-order logic have rules for adding and removing the quantifiers.

The first one:

$\forall x \varphi \vdash \varphi[x/a]$,

corresponds to the specialization principle : "what holds for all, holds for any".

And the same intuition supports the corresponding elimination rule:

if $Γ ⊢ \varphi[x/a]$, then $Γ ⊢ ∀x \varphi$, provided that $x$ is not free in $\Gamma$;

"if something holds for an arbitrary object, then it holds for all objects".

The first rule is quite intuitive, while the second one is a little bit tricky: the crucial point is the clause regarding "the arbitraty object" used to generalize.

Mauro ALLEGRANZA
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  • When you say the details of the semantics dictate how to read the formula, would it be correct then to complete separate the notion between the symbol/variable x for example, and any mathematical object that can be substituted in its place in the expression? I.e. is it correct to say a free variable is NOT a mathematical object, it is part of the formal language itself? – masiewpao Oct 25 '19 at 10:17
  • Also as a follow on, I am still somewhat confused about how to treat the bound variable? In proofs the way we treat the bound variable is as if it is an element of the set itself! By that I mean if we say in a proof let $x$ represents a real number, we use all the familiar operations of arithmetic and properties on 'x', even although $x$ is not actually an element of $\mathbb{R}$! It is an 'arbitrary' element of $\mathbb{R}$, but what this means in terms of how I would state what 'x' is is lost on me. – masiewpao Oct 25 '19 at 10:21
  • I think my knowledge of logic and formal languages my simply be too shallow; I will have to read up. Thank you very much! – masiewpao Oct 25 '19 at 10:26
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    @masiewpao - 1) YES; variables and constants and terms and formulas are all syntactical objects, i.e. are part of the (formalized) language that we use to speak of the world. The "world" (specifically, the mathematical part of it) is made of numbers, sets, functions, etc. Semantics is the way we link language and world: in formalized languages we have precise mathematical definition sof semantical concepts : interpretation, model, staisfaction, truth, etc. – Mauro ALLEGRANZA Oct 25 '19 at 11:11
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  • quantified variables are variables occurring into the scope of a quantifier : in $\forall x (x=0)$ the occurrence of car $x$ is bound while in $(x=0)$ it is free. Again, the syntactical rules of the Language allow us to detect in an unambiguous way what occurrences of a var are free and what bound.
    • – Mauro ALLEGRANZA Oct 25 '19 at 11:12
  • Ah OK, that makes sense to me. But this leaves me with one final clarification about the bounded variables (sorry for having to ask so many): In many examples in proofs we reason about a bounded variable by starting off with a statement such as "Let $x$ be an arbitrary element of $S$". From here we end up reasoning about $x$ just as we would a particular element of $S$. The thing I can't wrap my head around is that when we reason with a particular object, we can do so because we know its properties; and it is defined to be an element of some set. (cont) – masiewpao Oct 25 '19 at 12:24
  • For example if we say $2+2=4$, we know this is a true statement because of the properties of the elements of the set $\mathbb{Z}$. However, if $x$ i merely a syntactical object, how could we perform any reasoning on $x$ that we would for particular objects of a set? By this I mean since $x$ is a syntactic item, and not an element of some set, why then can we reason about $x$ as if it were an element of some set (giving it the associated properties of elements of that set)? I hope that makes sense, and thanks very much for explaining; I greatly appreciate the time you've spent. – masiewpao Oct 25 '19 at 12:27
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    @masiewpao - in order to evaluate the value of $x+2=4$ in e.g. $\mathbb Z$ we have to assign to $x$ a "temporary" meaning. This can be precisely specified through the semantics of the Language : $(x+2=4)[x \leftarrow 2]$ will be true while $(x+2=4)[x \leftarrow 1]$ will be false. – Mauro ALLEGRANZA Oct 25 '19 at 12:32
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    About free and bound variables. – Mauro ALLEGRANZA Oct 25 '19 at 12:40