In the way that they are used and understood in first order logic, a variable is nothing more than a piece of language. Variables helps us form mathematical sentences, so in a very literal sense, a variable is just a symbol, like the letters in a language. Variables are the core of formal mathematics, and they let us make general claims about what sorts of things are true.
A formula is a string of symbols in a formal language, one which is formed in accordance with the grammar of the language (usually called formation rules). For example, "$x\neq 3$" is a formula, because we understand the string to be asserting something, namely that "the object $x$ is not $3$". Without context though, the variable $x$ is sort of meaningless. It could be almost anything: a number, a set, a function, a person? Formally, $x$ is said to be a free variable of the formula "$x\neq 3$", because it has not been bound by a quantifier. Whether or not a variable is free or bound is dependent on the formula itself, it's not dependent on what other assertions are floating around; each formula categorizes its own variables.
Quantifiers are used in formal logic to bind a variable, and give it meaning by placing it in a context. For example, the sentence "$\forall x, x=3$" has a distinct and unambiguous meaning. It is saying "every object is $3$". This is false in basically every axiomatic system, but it is unambiguously false. We don't have to wonder about what $x$ is, because it's not really anything, it's just a formal way to say "object". We can also use different variables to talk about (possibly) different objects.
A sentence is a formula that has no free variables, they are typically formed by binding all the variables with quantifiers. A mathematical sentence has a totally unambiguous interpretation, it's either true or false. Whether a sentence is true or false depends on the surrounding axioms of the formal system. For example, the sentence "$\forall x, \exists y, y=x+1$" simply means "given any object ($\forall x,$), there is another object ($\exists y,$) that is $1$ more than the first ($y=x+1$)". If the objects we're talking about are integers, this is true, and it follows from the axioms of the integers.
Axioms are sentences which are implicitly assumed to be true. There a few ways to think of axioms. The platonic interpretation is that an axiom is something that is so simple and self evident that it absolutely must be true. Less rigidly, you could instead think of them as invisible premises, a way to not have to say "if so and so holds, then..." over and over. More intuitively though, axioms are what we use to describe what all these "objects" are. Each system of abstractions will have its own axioms, and mathematics is essentially the study of how to use axioms (simple truths) to get a thorough understanding of how an abstract system works.
A definition is a formal way to give a name to a specific object that you already know exists. That object could be a number, a function, a relation, etc, so long as you already know it exists. If we're being perfectly formal, the thing you're defining should almost always be a constant. For example, if you want to talk about the function that is associated with $y=3x+1$, what you really want to talk about is its graph, and what that graph (as an unchanging object) looks like. The symbols we use for definitions (like $3, e, \pi$) are sometimes the same as the symbols we use for variables. Because we only have so many symbols, that's unavoidable. When we use a variable to form a definition though, that symbol should stop being categorized as a variable, since it no longer serves the same linguistic purpose. For instance, it makes sense to say "$3\neq 4$", but it's absurd to write "$\forall 3, 3\neq 4$" because $3$ is a constant, not a variable.
The issue with variables in less formal approaches is that they tend to get merged with axioms and definitions in ways that are hard to untangle. For example, if we assert $y=3x+1$, what is this? If $x$ and $y$ are variables, then it can't be an axiom, because variables are supposed to be totally featureless elements of a language. Usually when something like this is done, the presenter is asking "if this is true, what else is true?" in which case it does actually amount to an axiom (or a premise, no difference). This is formally valid, but only if you stop considering $x,y$ to be variables; now they're just poorly defined constants. This is usually what people think about variables, a constant where you don't know what it is, but that's more akin to incompleteness. The formal system is unambiguously saying "this is a constant, not a variable", but it's not giving you enough information to figure out what the constant is actually supposed to be. This feature is useful when generalizing through deduction, but it is confounding to think about an "intentionally ambiguous constant", so people just call them variables (even though, technically, they are not).
Mathematical discourse, while introduces its own distinctions, abstracts away those insignificant for itself. Some of them surface in foundational and philosophical studies. Even then, several are of practical consequence, others are not.
Our usage of variables are mainly in three ways:
- As a symbol representing an unknown value.
- As a symbol representing a range of values.
- As a place-holder.
All three senses share the core idea that the symbol as the variable has no intrinsic semantic value; we shall focus on the second and the third ones for the present question.
The distinction between (2) and (3) can be illustrated as follows:
Suppose we calculate $y$ such that $y = x + 5$ as $x$ takes integral values in the range of $\{1, 2,\ldots, 10\}$. We may view $x$ as assigned to the number $1$ so that $x$ refers to $1$ and $y$ refers to $6$, and so on. Thus, what $x$ refers to varies with the range of the values.
Alternatively, we may view $x$ as a place-holder that marks a place in the expression. Thus, $x$ refers to the dots (the gap) in $y =\ldots + 5$ to be filled in by appropriate symbols. We do not actually assign values to $x$ to refer to, but substitute numbers (in fact, numerals, but we skip over such details) from the set $\{1, 2,\ldots, 10\}$ for $x$ and those substitutes are the values of the variable $x$ as the author states. We write the numbers in the place to which $x$ points at (i.e., the dots). It might be helpful to compare this sense of variable to place-holder as the linguistic term. Consider the sentence:
“It is a conjecture that $n+1, n+2,\ldots, n+k$ being composite numbers, there are $k$ distinct primes $p_{i}$ such that $p_{i}$ divides $n+i$ for $1\leq i\leq k$.”
The occurrence of ‘It’ in the preceding sentence is not the genuine subject of the sentence; it is a placeholder.
In the usual mathematical parlance, these two perspectives is conflated. It is up to one's cognitive choice to shift from one viewpoint to the other. If the set $x$ ranges over is large, one might well conceive of the place-holder $x$ “as if” $x$ were substituted by the specified values in abstracto. Since we fix a set $\{1, 2,\ldots, 10\}$ in both cases and obtain the set $\{6, 7,\ldots, 15\}$, it does not matter which one we choose. Thus, the terms ‘variable’ and ‘place-holder’ are held as synonymous.
Likewise, in the sentence $\forall x(x\in\mathbb{R}\rightarrow P(x))$, $x$ is taken with varying referents. However, when we instantiate it to a constant in the language of logic, say $\alpha$ and so $P(a)$, $x$ is taken with varying occurrence (i.e., its occurrence is replaced with the occurrence of $\alpha$).
Indeed, the distinction perspectives can be translated into substitutional interpretation of quantifiers as contrasted to the familiar objectual interpretation and thus the truth-conditions of quantified sentences can be differentiated in logic. Let us briefly touch on this.
The usual interpretation of the standard quantifiers $\forall$ and $\exists$ is objectual; values are regarded as ‘objects’ to which the variables refer. This requires us to fix a domain of values, otherwise, the variables cannot refer or refer to $\emptyset$.
If a quantifier is interpreted substitutionally, appropriate expressions are substituted for the variable, and the resultant sentence is evaluated to true or false. Notice that substitutional quantifiers do not require us to fix a domain to provide the referents of the variables, for the variables actually do not refer, but leave their places.
The usual symbols for substitutional universal and existential quantifiers are $\Pi$ and $\Sigma$, but I think it is better to denote them by $\overline{\forall}$ and $\overline{\exists}$ in order both to reserve the former symbols for their other uses and to facilitate the association with the familiar quantifiers, given the contemporary ease we have for typographical variations. So, let us consider the sentences:
- $\exists x(x\text{ is a horse})$
- $\overline{\exists}x(x\text{ is a horse})$
and their instantiation to ‘Pegasus is a horse’.
From the received point of view, the sentence ‘Pegasus is a horse’ is false, because the sentence (1) seeks its referent in the domain of horses of which Pegasus is not a member. But from the substitutional point of view, it is true, because the resultant sentence expresses a cultural fact independently of a domain of discourse to seek a referent for Pegasus. But ‘the table is a horse’ would be false substitutionally as it would be objectually. Thus, we make sense of the author's use of the phrase “significant substitute”: ‘Pegasus’ is a significant substitute, but ‘the table’ is not.
For those interested, I recommend Gabriel Uzquiano’s SEoP article Quantifiers and Quantification to gain more insight about the topic.
To conclude, we sometimes imply the referents of a variable, and sometimes, its occurrences. We call both the referents and the substitutes (of occurrences) the values of a variable. We have to take caution: Taking the former sense as the sole one may cause confusion when the latter one is also used in the context.
Best Answer
The terminology for these placeholders is informal, and usage is guided by context and framing or intended signalling. In this example, the parameter $k:=gMm$ generally varies but is fixed in the context of the inverse proportionality of $F$ and $d^2.$ In this example, $m_1,$ which equals $P,$ instantiates $m,$ but the three objects have somewhat different ontological statuses.