Are variables mathematical objects?

[HyperbolicMan]

Just of out of curiosity, is it possible to rigorously define the notions of variable and constant? It seems to me that if these notions don't have rigorous definitions, then our way of thinking about mathematics pretty much falls apart.

 

[chiro]

Just of out of curiosity, is it possible to rigorously define the notions of variable and constant? It seems to me that if these notions don't have rigorous definitions, then our way of thinking about mathematics pretty much falls apart.

Lets for the moment restrict variables to real or complex numbers.

A constant basically has no freedom: that is, it has no variation and needs no parameter to describe it (it is dimensionless).

On the other hand a variable (a number in this case) has a level of variation, and to accurately describe a system with this variation, you need to know the value of the parameter at a particular point to properly define the system.

An example is the equation y = x + 2. You have to give a value of x to get a value of y, but to describe the system (which is just a line), you have to supply what value of x you want to use to get your corresponding y.

Mathematics however, is not restricted to objects as simple as numbers.

You can have things like vectors and matrices, obscure groups and so on.

So to sum up, constants in any form of a mathematical object are dimension-less and anything that is not zero-dimensional has variation of some sort.

 

[honestrosewater]

I think yes and no because you need to make a distinction.

Reason for no: Variables and constants are objects of the languages used by mathematicians to talk about mathematical objects. Variables and constants are linguistic objects in the same sense that lines and figures are geometric objects and numbers and operations on them are numeric objects. Variables and constants are objects that are made from alphabets, sit in strings, and refer to other objects. When I say "let x be a real number", the role of "x" as a variable representing a number does not make it a number any more than the phrase "a real number" is a number. They are both pieces of a language. They represent numbers. Likewise, the constants/names "1" and "one" are not numbers but representations of numbers. This distinction can be brought out more clearly by noting that there are no such things as variables and constants in, say, number theory, group theory, topology, or set theory. There are sets, points, numbers, group operations, etc. Addition does not operate on constants or variables but on numbers. In this sense, no, the variables and constants are not part of the mathematical theory being studied.

Reason for yes: Languages are studied mathematically. When a language is being studied as a mathematical object, the variables and constants of the languages are being considered as mathematical objects. You just have to take care in such situations to distinguish between the different types of objects that you're dealing with: the objects being referred to by the object language, the object language itself, the representation of the object language in your metalanguage, the representation of the metalanguage, etc. And yes, you can define formal languages and interpretations for them where variables and constants are as precisely defined as any other mathematical object and reasoned about in rigorous ways. Lots of people work with and study formal languages to various extents, people working in formal/mathematical logic, model theory, proof theory, computer science, linguistics, and perhaps philosophy of language and others.

What is the reason for your concern? How would you define variables and constants?

 

[JeremyHorne]

Somewhat of an elaboration on honestrosewater:

A formalization of set theory refers to variables as object language entities. That is, the formalization is a metalanguage talking about things - object language -like symbols (including those for variables and constants) and expressions. This is like talking about Spanish, such as spelling, grammar, alphabet, etc. in the English language, the former being the metalanguage, and Spanish being the object language. Note, however, that the object language can become the metalanguage. Spanish could be the metalanguage talking about English. However, in logic, we have type theories that delineate levels to avoid situations like using the logic as an object language in one moment and a metalanguage in another or using the language to talk about itself. A distinction between object and metalanguage is made by the symbols used.

Further research can proceed along the lines of distinguishing between semantics and syntax.

A further line of inquiry is philosophical concerning the notion and problems of representation, starting with Plato's Republic - cave analogy and divided line.

 

[HyperbolicMan]

What is the reason for your concern? How would you define variables and constants?

The thought came to me while working with a function of the variable t. I was trying to figure out what its graph looked like. I was thinking about t varying over the real numbers and it occurred to me that this notion of 'varying' is kind of vague. So I thought about it, and I realized that, with algebra, we generalize from working with specific objects (i.e. 1,2,3) to working with all objects in a set by means of variables (i.e. x). However, this 'leap' is never proven. We go ahead working with indefinite objects as though they were well defined. For example, suppose I said "Let x be a real number. Then, (x+1)^2=x^2+2x+1." Because I worked with the variable x, I can make the claim that this statement will hold whenever I substitute a real number in for x. I've always just taken this as intuitively obvious (and I still do), but I wonder how math/logic handles this issue.

If, for some awful reason, it turned out that working with a variable DOESNT prove a statement over all of a domain, then we'd have to rethink the way we do mathematics, right?

 

[honestrosewater]

Well, you want to know about quantification and scope, I think. Logic handles variables with quantifiers. Standard logic has two quantifiers, universal (for all) and existential (there exists). And logics with quantifiers have rules for dealing with formulas that include these quantifiers, just like there are rules for dealing with formulas that include "and", "or", and "not". You don't always mention the quantifiers explicitly or consistently in normal math talk. When you say "Let x be a real number. Then, (x+1)^2=x^2+2x+1," the "x" is quantified universally. You could also have said "For all x, if x is a real number, then (x+1)^2=x^2+2x+1". And when you prove a statement of that form, you just use the rules for quantifiers. Does that start to address what you were thinking about?

 

[AlephZero]

Because I worked with the variable x, I can make the claim that this statement will hold whenever I substitute a real number in for x. I've always just taken this as intuitively obvious (and I still do), but I wonder how math/logic handles this issue.

This is a fundamental axiom of logic. Expressing it in words rather than symbolism,

If "x is a member of set S" implies that "x has property P", then "every member of S has property P".

This is completely obvious for finite sets, and almost as obvious for infinite sets, but it can't be "proved" except by assuming a different axiom that is equivalent to it.

Fundamental ideas about sets like "x is a member of S", "for all members of S", "for each member of S", etc, can not be "defined" any more than ain idea like "a point" can be defined in geometry. What mathematicians do is make a complete list of the rules (or axioms) that you can use to work with the ideas.

When you start learning math, actually you are starting somewhere "in the middle" of the subject, not at the beginning. Ideas like "numbers", "variables", etc are not formally defined until after you have got a lot of practice in actually using them (and that is what you are currently learning in your math classes, of course).

 

[HyperbolicMan]

Ahhh it seems like I should have been asking about universal quantification all along. If I'm following correctly, honestrosewater and alephzero, sets are defined by properties. Proving a statement for an arbitrary element of a set implies that the statement is true for all elements of the set. And vice-versa, I'm guessing. Now (and sorry if this is falling into the realm of philosophy) I'm wondering: How do we know that an object "x" is truly arbitrary? (x+1)^2=x^2+2x+1 where x is a real number. Does x actually stand for a SPECIFIC real number throughout the entire problem? In other words, is x always equal to some value that we may or may not be able to determine? My reason for thinking this is: 2x+1=3 => x=1. Here, x stood for 1 ever since we started working the problem. However, x^1-1=0 => x=1 or x=-1. The specific number that x equals is either 1 or -1. We'll never be able to figure out which one. I am referring to x as though it is not quantified. Please let me know if I'm not being clear!

Thanks for all the responses so far!

 

[honestrosewater]

If AlephZero's response is helpful to you in some way, great. But I have to say that, if taking the usual meanings of these words in the areas being discussed, it is technically incorrect in several ways.

Axioms of logics do not talk about sets. They talk about formulas and truth values. The statement

If "x is a member of set S" implies that "x has property P", then "every member of S has property P".​

doesn't make much sense to me, but I suppose it is saying that, if you prove that some arbitrary member of a set has a property, then you can conclude that every member of the set has that property. If this is true, it will be true because of how sets and set membership are defined.

A set membership relation, universal quantification, and existential quantification can indeed be well-defined in terms of other things. You probably can come up with a perfectly satisfactory definition of points in terms of other geometric objects also. In any particular theory (or particular axiomatization of a theory, depending on how you slice things), you will have undefined terms, yes, because you have to start somewhere. But that doesn't mean that you always have to start at the same place or with the same things. And even the notion of your starting objects being undefined is a little misleading because the whole theory serves to define the objects in its domain. If you want to know what a set is, it's a member of the domain of a model of a set theory. The whole theory is talking about sets and exactly what properties they can have, so it's not like they're some kind of mysterious object that you use and sweep under the rug.

If I'm following correctly, honestrosewater and alephzero, sets are defined by properties.

Letting sets be defined by properties will get you into a kind of trouble that might not trouble you, but if you are curious, ask about what's wrong with unrestricted comprehension in set theory. If you want to target people who know about these things, don't mention Russell's paradox.

In practice, you can define a set in three ways:
(i) name its members (only works for finite sets obviously)
(ii) define some property that all of its members satisfy (must be careful here)
(iii) name at least one member of the set and give a recursive rule for deriving the other members from the one(s) named.

Proving a statement for an arbitrary element of a set implies that the statement is true for all elements of the set. And vice-versa, I'm guessing.

Yes, by the definition of what it means for an element to be arbitrary. You cannot give an arbitrary element of set S any properties by which is can be distinguished from other members of S. If I said "let x be an even number", then it would be an arbitrary even number but not an arbitrary number because not all numbers have the property of being even. Does this distinction make sense?

Now (and sorry if this is falling into the realm of philosophy) I'm wondering: How do we know that an object "x" is truly arbitrary? (x+1)^2=x^2+2x+1 where x is a real number. Does x actually stand for a SPECIFIC real number throughout the entire problem? In other words, is x always equal to some value that we may or may not be able to determine? My reason for thinking this is: 2x+1=3 => x=1. Here, x stood for 1 ever since we started working the problem. However, x^1-1=0 => x=1 or x=-1. The specific number that x equals is either 1 or -1. We'll never be able to figure out which one. I am referring to x as though it is not quantified.

The statements you list don't have to be true. You are assigning values to the variables to make the statements true. If those were taken as just FOL formulas, the variables are free, and if you assign "2" to the "x" in "2x+1=3", the statement is false. There is no real number that you can assign to the variable in the first to make it false, and this fact is what is captured by the universally-quantified version being true. Loosely, whether a variable stands for a specific value depends on the variable's quantification and the values it ranges over. But we haven't said very clearly what we mean by a variable standing for a value, so that is only a start to the truth.

I would say it is closer to the truth to say that the free "x"s all stand for some member of whatever set you said they ranged over. This is because, when you interpret the formulas, they will be assigned a single value from that set. That only certain assignments of values will make the statements true doesn't change the role of the variable. It's something about the statement as a whole. In "2x+1=3", "x" doesn't stand for 1 in the same way that "1" stands for 1. "x" doesn't really stand for 1 at all. 1 is in its range and so can be assigned to it. This assignment happens to make the statement true and happens to be the only assignment from that set that does so. But as far as "x" is concerned, 1 is just another value in its range.

 

[CompuChip]

2x+1=3 => x=1
[...]
I am referring to x as though it is not quantified. Please let me know if I'm not being clear!

To follow up on honestrosewater's reply to this last part: what you actually were saying was:
For all x in your universe (e.g. real numbers), if the expression 2x + 1 = 3 is true, then you can infer that x has the value one.
So x is implicitly quantified.

Note by the way, it does not say anything about the case where "2x + 1 = 3" is false, or about the case where "x = 1". This may be a bit confusing, because in this case, "2x + 1 = 3" does not just imply (=>) that x = 1, but it is logically equivalent (<=>) to it. Another example would be "x = 2 => x² = 4". So "for all values of x, if x has the value 2, then its square has the value 4" (the quantification is still there, although it's a bit pointless). Note that nothing is said about the case where x does not have the value 2 (x² may or may not equal 4), and it also does not state that if x² = 4 then x must necessarily be 2.

The use of x in both expressions does mean that once you assign a value to x, it must be the same everywhere.

Usually, when you do not say what "x" is but just use it as a variable, you are using universal ("for all") quantification. An example of the other kind, existential ("there exist(s)") quantification, would be the statement: "2x + 1 = 3 has a solution."
If you would express that formally, it would be something like:
"There exist(s) (a) value(s) of x (in some specified universe, e.g. real numbers), such that if you assign them to x the equation 2x + 1 = 3 becomes a true statement.".

 

[HyperbolicMan]

Thanks for the imput guys! I think I thought of an easy way to get my point across:

Let x be a real number. What is the truth value of a statement such as "x=3" ?

Based on what u guys have been saying, I'm going to guess that this is undefined. But it tackles the idea about a variable standing for some specific but 'unknowable' value (and therefore arbitrary). Note, I am referring to x as a single real number without universal quantification. I understand that if x were universally quantified, then the whole idea of x being a specific thing meaningless because we're considering all possible values of x. But think this might be important because aren't the two approaches equivalent by universal instantiation/generalization?

 

[mXSCNT]

Variables are the parameters of functions. They describe how the function varies with respect to a given dimension. This is compatible with the logical-quantifier view (in some formulations of logic, everything is a function), but is intuitively clearer, in my opinion. When considered as the parameters of functions, variables are more easily seen as mathematically "real".

 

[EnumaElish]

In some sense any constant identifies a subset of a space. For example, f(x) = x+1 is f(x,y) = x+y constrained to the y = 1 plane. In that sense any constant can be seen as a special value for a variable in a higher-dimensional space and therefore the distinction between a constant and a variable as a distinction between a subset of a space (a plane, or a subspace) versus the space itself.

 

[honestrosewater]

Thanks for the imput guys! I think I thought of an easy way to get my point across:

Let x be a real number. What is the truth value of a statement such as "x=3" ?

Based on what u guys have been saying, I'm going to guess that this is undefined.

If x is an arbitrary real number, is x necessarily 3? No. Is x possibly 3? Yes. This is not a difficult question if you think about it. Perhaps you want to hear that the truth value is variable. It depends on the value assigned to x. The truth value of a formula containing variables is a function of an assignment of values to the variables. So you need to say "the truth value under this assignment...".

Note, I am referring to x as a single real number without universal quantification.

But you're not. You said it was a (arbitrary) real number: x now ranges over the set of real numbers. The difference between a universally quantified bound variable and an arbitrary member of a set is a technical/syntactic one. They represent the same concept. That x is a variable is saying something about how it works with assignments. Each assignment assigns it a single value, but it does not need to be assigned the same value by every assignment. That is the role of constants.

 

[disregardthat]

If you by mathematical object mean e.g. the kind of object a "countable set" is in set theory, or what "point" or "line" is in euclidean geometry, then variables are not mathematical objects, at least not in its usual sense. A variable is not anything, it is used as a way of defining a rule. When we are defining a function, say f on the reals by f(x) = 2x+3, we mustn't think of x as being anything. In principle we are stating a rule:

"If x is a real number, then f(x) = 2x+3".

The prior form of definition tricks us into believing "f(x)" makes sense formally without defining x. Similarly "2x+3" makes no sense without a prior definition of x. Note that this is different to the case where we say x is an indeterminate, indeterminates have well-defined set-theoretic models. Nevertheless, only "f(2)", "f(0)" and "f(e)" and so on makes sense. The rule can be only applied whenever we define x. E.g. let us say x = 5. Then the rule can be applied since x is a real number, and f(x) = 2x+3=13. "x is a real number" is meaningless prior to a definition of x.