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Are there precise definitions for what a variable, a symbol, a name, an indeterminate, a meta-variable, and a parameter are?

In informal mathematics, they are used in a variety of ways, and often in incompatible ways. But one nevertheless gets the feeling (when reading mathematicians who are very precise) that many of these terms have subtly different semantics.

For example, an 'indeterminate' is almost always a 'dummy' in the sense that the meaning of a sentence in which it occurs is not changed in any if that indeterminate is replaced by a fresh 'name' ($\alpha$-equivalence). A parameter is usually meant to represent an arbitrary (but fixed) value of a particular 'domain'; in practice, one frequently does case-analysis over parameters when solving a parametric problem. And while a parameter is meant to represent a value, an 'indeterminate' usually does not represent anything -- unlike a variable, which is usually a placeholder for a value. But variables and parameters are nevertheless qualitatively different.

The above 2 paragraphs are meant to make the intent of my question (the first sentence of this post) more precise. I am looking for answers of the form "an X denotes a Y".

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@Jacques, such philosophical questions are perfect as community wiki. –  Wadim Zudilin Jun 25 '10 at 7:24
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@Wadim: I was rather hoping that this was not really philosophical (anymore). Isn't this so basic that there should be definitions? –  Jacques Carette Jun 25 '10 at 11:36
    
@Jacques: this is the type of thing that is so basic that there are not mathematical definitions. Unless mathematicians were proving things about variables, why would they need to define them precisely? –  Carl Mummert Apr 27 '11 at 10:14
    
@Carl: This is exactly what foundation studies are all about: take things that were thought basic and define them formally. Both logic and set theory did that, for a part of mathematics. But it appears that there are still parts of mathematics which are not formal. Why is that? –  Jacques Carette Apr 27 '11 at 12:30
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6 Answers 6

In written English (and of course other languages), we have linguistic constructs which tell the reader how to approach the ideas that are about to be presented. For example, if I begin a sentence with "However, . . .", the reader expects a caution about a previously stated proposition, but if I begin the sentence with "Indeed, . . . ", the reader expects supporting evidence for a position. Of course we could completely discard such language and the same ideas would be communicated, but at much greater effort. I regard the words "variable", "constant", "parameter", and so on, in much the same way I regard "however", "indeed", and "of course"; these words are informing me about potential ways to envision the objects I am learning about. For example, when I read that "$x$ is a variable", I regard $x$ as able to engage in movement; it can float about the set it is defined upon. But if $c$ is an element of the same set, I regard it as nailed down; "for each" is the appropriate quantifier for the letter $c$. And when (say) $\xi$ is a parameter, then I envision an uncountable set of objects generated by $\xi$, but $\xi$ itself cannot engage in movement. Finally, when an object is referred to as a symbol, then I regard its ontological status as in doubt until further proof is given. Such as: "Let the symbol '$Lv$' denote the limit of the sequence $\lbrace L_{n}v \rbrace_{n=1}^{\infty}$ for each $v \in V$. With this definition, we can regard $L$ as a function defined on $V$. . . "

So in short, I regard constructing precise mathematical definitions for these terms as equivalent to getting everyone to have the same mental visions of abstract objects.

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Excellent! Your discourse above is very reminiscent (to me) of the same discourse present in Leibniz's work, and later Frege as well as Russell, which served to really clarify the mathematical vernacular. The informal use of words led to some formalizations (think set theory and logic) which really help put mathematics on a much more solid foundation than before. This is now established canon. But why do other words in common mathematical usage (with mathematical meaning) escape this treatment? –  Jacques Carette Apr 27 '11 at 12:36
    
I believe that "variable", "constant", and "parameter" have identical set theoretic meaning, as they operate as adjectives describing elements of sets within any given proof, and the validity of proofs depends only on the properties of the elements of the sets under consideration, not the adjectives used to describe the elements. So though we regard "variable" as a noun, it arises from the mental abstraction of an adjective. Objects which seem to be amenable to precise mathematical definitions seem to arise as abstractions of nouns. (That's the best answer I can come up with unfortunately!) –  user14717 Apr 28 '11 at 3:02
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Regarding the status of variables, you probably want to look at Chung-Kil Hur's PhD thesis "Categorical equational systems: algebraic models and equational reasoning". Roughly speaking, he extends the notion of formal (as in formal polynomials) to signatures with binding structure and equations. He was a student of Fiore's, and I think they've been interested in giving better models (inspired by the nominal sets approach) to things like higher-order abstract syntax. I've been meaning to read his thesis for a while, to see if his treatment of variables can suggest techniques that could be used for writing reflective decision procedures which work over formulas with quantifiers.

For schematic variables or metavariables, there's a formal treatment of them in MJ Gabbay's (excellently-titled) paper "One and a Halfth-Order Logic"

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I've been reading other of Gabbay's papers, which I have been greatly enjoying. I have tried several times to erad Fiore's work, but my understanding of CT is just not strong enough to cope. It appears quite unfortunate, since he does seem to have a lot to say about questions I have been asking myself. –  Jacques Carette Jun 24 '10 at 20:09
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Intriguing question...

If there are definitions then as far as I know they're pretty much unspoken ones. Maybe someone has actually codified them somewhere, but I'm guessing not- so I'm going to take a few guesses and stick this answer as community wiki in a bid to get some kind of consensus:

I can (fairly confidently) vouch for

Variable: The argument of a function (sometimes a truth function :))

Indeterminate: Dummy variable used to prove statements with universal quatifiers

Parameter: A numerical variable determining an object

I would take guesses at:

Symbol: A function or functional (ie. more complex than simply an object) that is a variable

Name: The argument of a truth function

And I have no idea about:

Metavariable: ?

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Feel free to mess about with this ^^^^^^^^ –  Tom Boardman Jun 24 '10 at 19:46
    
This is indeed what I was looking for. Some (like indeterminate), seem quite close to the mark. I am less keen on your characterization of 'variable' though. Don't variables sometimes occur outside the scope of a function? [Although another might be that they don't and those who use variable in that context are guilty of either sloppiness or misunderstanding of what a variable is]. –  Jacques Carette Jun 24 '10 at 20:06
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Of the various types of "placeholder", certainly a couple have definite mathematical meanings. In logic, the meaning of free and bound variables is set out in detail. And I take "indeterminate" to be a term used with a precise meaning in algebra; in polynomial rings, for example, the indeterminates are not exactly independent variables in the conventional sense of functional notation.

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I know this - I could have put all of that in my question (which was long enough as it is). What I am seeking is exactly some of those precise meanings. I do know the precise meanings from logic (I did mention $\alpha$-equivalence, right?). But the meaning of a term in logic is not always the meaning of that term in the rest of mathematics... –  Jacques Carette Jun 24 '10 at 19:10
    
It might be clearer to ask first where the dominant idea of "function", as mathematicians now understand it, is not the only useful one. And then ask for the descriptive terms to be clarified. –  Charles Matthews Jun 24 '10 at 21:01
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It seems to me that this PhD thesis may well contain answers that I find satisfying. The discussion on p.52 is particularly appealing, but the whole thesis is strewn with similar passages discussing mathematical terms which are frequently left (formally) undefined in the mathematical literature.

Warning: many people who post here would likely call this thesis part mathematical philosophy and part computer science, and find little modern mathematics in it. But then again, as mathematicians seem to be trying to take type theory back for their own, maybe this kind of work will come back in vogue too.

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I used to worry a lot about the “ontological status of variables”, and I was eventually able to achieve a modicum of ontological security, at least, with respect to the simple domains that interested me, by taking a pattern-theoretic view of variables. In this view, you shift the question from the status of an isolated variable name like “$x$” to the syntactic entity “$S \ldots x \ldots$” in which the variable name occurs. You may now view “$S \ldots x \ldots$” as a name denoting the objects denoted by its various substitution instances.

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You're telling me that variables are different when viewed intensionally or extensionally. We agree. So, can you formalize not just variable, but each of the terms I gave, with an explicit denotation? –  Jacques Carette Jun 24 '10 at 19:13
    
@ Jacques Carette –– That would require an excursion into details that experience tells me are not likely to be tolerated here. Shrift as shortly as possible, it helps to have the classical notion of general or plural denotation, which classical thinkers deployed to good effect long before they had classes. –  Jon Awbrey Jun 24 '10 at 19:28
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