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I was just wondering about the formal definition of equation, I mean in terms of logic and the theory of sets. Suppose for example I wanted to define an equation on $\mathbb{R}$, of course it might be anything instead. I know every number is a set, and also I know that equality of sets has been defined previously. So I often hear people say something like "A linear equation is an object of the form $a_{1}x_{1}+a_{2}x_{2}+ \cdots + a_{n}x_{n}=b$". But how do you justify the part of "$x_{i}$ is an unknown". Because in that case I would also say that $3(2)+4(5)=0 $ is an equation.

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By quantifying (universal) over things in $\mathbb R$ (one quantifier for each $x_i$) maybe? – Forever Mozart Feb 11 '13 at 21:39
The possibilities depend a bit on how your language is constructed, and the notion of an equation may be implicit in the way that "=" is defined in that language. You may also want to consider what sense you might want to make of "equations" like $x^2=2$ and $x^2=-1$ - are they allowed to be equations or not? Some languages have explicit terms which can function as "unknowns" specifically because they are constructed to allow expressions such as you give for a linear equation. – Mark Bennet Feb 11 '13 at 21:57
I think Berci's and DoubleTrouble's answers are good ones in terms of defining "equation" just like mathematicians do by using the rules of logic (that was in fact what I was trying to mean in my question). But Makhlom's answer is more of philosophical treatment and his point is on the formal definition of "equation in general", as a component of the rules of logic, very nice conception. So I chose Makholm's answer as the best. – Daniela Diaz Feb 12 '13 at 19:20
up vote 4 down vote accepted

You won't find a formal definition of "equation" and "unknown", because they are partially informal concepts. The formula on the paper is what it is: two expressions, usually containing variables, joined by an equals sign. But there's no fixed rule that will tell you, from looking at the formula, which of the variables (if any) are unknowns.

Being an "unknown" is not a property of a variable or a formula. It expresses something about what you, the human mathematician looking at it intend to do with the equation. The roles of unknowns and constants can change simply by virtue of you changing your mind.

For example, if we're looking at the formula $$ x^2+y^2=1 $$ we can choose to "solve for $x$", that is, treat the variable $x$ as an unknown and everything else as constants -- and we'd end up with $x=\pm\sqrt{1-y^2}$ -- or we could have made a different choice end with $y=\pm\sqrt{1-x^2}$ instead. Neither approach to the formula is objectively right or wrong. One of them may lead us closer to whatever our eventual goal is, but the formula itself doesn't know what that eventual goal is.

In particular when we say "treat $x$ as an unknown", there is no implied "... even though it really isn't". Nothing is an unknown in and of itself; variables are unknowns when and if we choose to treat them like one.

Also note that the formal rules of what one is allowed to do to a formula doesn't distinguish between knowns and unknowns. We can always replace $(p+q)^2$ with $p^2+q^2+pq+qp$, no matter whether $p$ and $q$ are knowns, unknowns, or some complex things built from several of each.

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I don't really get what an informal concept it. If it isn't a mathematical concept, then it must be a primitive concept. – Git Gud Feb 11 '13 at 22:09
@GitGud: Your usage of at least one of those terms must be very strange. Do you think "papal conclave" is a "primitive concept"? It certainly isn't a mathematical one. – Henning Makholm Feb 11 '13 at 22:15
Primitive concepts depend on which theory you're working on. If you can't (or choose not to) define something, then, after transmitting an idea of what you want that thing to mean, it becomes a primitive concept with respect to that theory. – Git Gud Feb 11 '13 at 22:20
@DanielaDiaz: Yes, there is a formal distinction in logic between "variable symbols" and "constant symbols". However, according to that distinction, "constant symbols" are things like "0" and "7", whereas things like "$a$" and "$x$" are both variable symbols. In $ax^2+bx+c=0$ the constant symbols are 0 and 2, whereas $a,b,c,x$ are all variables. When we solve the quadratic equation we treat $x$ differently from $a$, $b$, and $c$, but the difference is in our choices, not in any formal distinction between those four variables. (...) – Henning Makholm Feb 11 '13 at 22:43
(...) From a formal logic standpoint we can express the difference as "we treat $a, b, c$ as if they were constant symbols", which works because everything the rules of logic allows us to do to a constant symbol they also allow us to do to a variable. But this difference is still one of intent, not of something that exists at the raw level of formal proofs. – Henning Makholm Feb 11 '13 at 22:45

I would pick a set $x$ which is different from all the numbers we want to use, and regard it as an unknown in the formalism. Then, for an $n\in\Bbb N$, $\ x_n:=\langle x,n\rangle$ (ordered pair).

Then, define terms as finite sequences, using these variables, perhaps the numbers, and the operation symbols (for these, a new, so far unused sets can be used).

Then, an equation would be an ordered pair $\langle\tau,\sigma\rangle$ of terms. Its meaning is defined then straightforwardly.

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I would say that an equation is a logical predicate. That is, an open dictum whos truth value is determined by some input e.g.

$$P(x) \ : \ x + 2 = 4$$

where $x$ belongs to some domain. To solve an equation is to find the (possibly infinitely long) truth table for this predicate.

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Well, yes, except that some predicates are not equations. For example $Q(x) \equiv x>3$ is not an equation. And most times when we solve equations that are no "truth tables" involved. What a solution means here is just a simpler description of the solution set, where the meaning of "simpler" depends on context and intentions. – Henning Makholm Feb 11 '13 at 22:26
I didn't say that all predicates are equations. Just that an equation can be considered to be a predicate. OP asked for the formal logical explaination of an equation and what it means to solve it. I don't believe he asked for simplification. – DoubleTrouble Feb 11 '13 at 22:45
However talking about "find the infintely long truth table" is a simplification of what we do when we solve equations. What we actually do is to find a description of the solution set that is simpler and/or easier to work with than the description provided by the equation itself. – Henning Makholm Feb 11 '13 at 22:54
x @Double: Exactly. From a formal logical point of view, simply writing, say, $$\{ x\mid 7x^5+x^4-22x^3+8872x-1=0\}$$ is a perfectly good construction of that solution set. So what we're doing when solving the equation is not "finding the solution set" or "finding the truth table", because logically we already have it by a few strokes of the pen. The task is to find a description of the solution set that is easier to work with (for the purposes at hand) than the one above. And that is not a question of formal logic, but of extra-logical intent. – Henning Makholm Feb 11 '13 at 23:05
@DanielaDiaz: (This seems to belong more in the thread under my answer, or am I misunderstanding you?) There's nothing wrong with defining "linear equation" in this way and then define what you want "unknowns" to mean in the context of that definition. What I'm saying is that there's no formal definition of "unknown" in general (that is, without the restriction to unknowns of a linear equation in canonical form) which matches the way mathematicians use the word in practice. – Henning Makholm Feb 12 '13 at 18:01

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